![EDITORIAL COMMITTEE
Michael Loss, Managing Editor
John Etnyre Angela Gibney Catherine Yan
2020 Mathematics Subject Classification. Primary 16-XX, 16N40, 16S34, 16S36, 11T71,
94BXX, 94B05, 94B15, 94B60.
Library of Congress Cataloging-in-Publication Data
Names: Rizvi, S. Tariq, honoree. | Leroy, André (André Gerard), 1955– editor. | Jain, S. K.
(Surender Kumar), 1938– editor. | International Conference on Noncommutative Rings and
Their Applications (7th : 2021 : Lens, France), author. Virtual Conference on Quadratic
Forms, Rings and Codes (2021 : Lens, France)
Title: Algebra and coding theory : virtual conference in honour of Tariq Rizvi, Noncommutative
Rings and Their Applications, VII, July 5–7, 2021, Université d’Artois, Lens, France : Virtual
Conference on Quadratic Forms, Rings and Codes, July 8, 2021, Université d’Artois, Lens,
France / A. Leroy, S.K. Jain, editors.
Description: Providence, Rhode Island : American Mathematical Society, [2023] | Series: Con-
temporary mathematics, 0271-4132 ; volume 785 | Includes bibliographical references.
Identifiers: LCCN 2022048347 | ISBN 9781470468590 (paperback) | ISBN 9781470472986 (ebook)
Subjects: LCSH: Coding theory–Congresses. | Algebra–Congresses. | AMS: Associative rings
and algebras. | Associative rings and algebras – Radicals and radical properties of rings –
Nil and nilpotent radicals, sets, ideals, rings. | Associative rings and algebras – Rings and
algebras arising under various constructions – Group rings. | Associative rings and algebras –
Rings and algebras arising under various constructions – Ordinary and skew polynomial rings
and semigroup rings. | Number theory – Finite fields and commutative rings (number-theoretic
aspects) – Algebraic coding theory; cryptography. | Information and communication, circuits –
Theory of error-correcting codes and error-detecting codes. | Information and communication,
circuits – Theory of error-correcting codes and error-detecting codes – Linear codes, general. |
Information and communication, circuits – Theory of error-correcting codes and error-detecting
codes – Cyclic codes. | Information and communication, circuits – Theory of error-correcting
codes and error-detecting codes – Other types of codes.
Classification: LCC QA268 .A388 2023 | DDC 512/.4601154–dc23/eng20230415
LC record available at https://lccn.loc.gov/2022048347
Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)
DOI: https://doi.org/10.1090/conm/785
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![Contemporary Mathematics
Volume 785, 2023
https://doi.org/10.1090/conm/785/15770
A note on checkable codes over Frobenius
and quasi-Frobenius rings
R. R. de Araujo, A. L. M. Pereira, and C. Polcino Milies
Abstract. We study checkable codes in finite Frobenius and quasi-Frobenius
rings and show that the Jacobson radical of a quasi-Frobenius ring is checkable
if and only if the ring is Frobenius. Also, we show that in a quasi-Frobenius
ring R every maximal left ideal is checkable and that R is code-checkable if
and only if R is a principal right ideal ring.
Dedicated to Professor Syed Tariq Rizvi for his contribution to Ring and Mod-
ule Theory.
1. Introduction
Cyclic codes over a finite field F were introduced by Prange [14] in 1957 and
can be realized either as ideals in a factor ring F[X]/Xn
− 1 or in its isomorphic
image FCn, where Cn denotes a cyclic group of order n.
Extending this idea, S.D. Berman [2] in 1967 and F.J. MacWilliams [13] in
1970, independently, introduced the notion of a group code, which can be defined
as a left (or right) ideal in a finite group algebra.
In 2008 J.A. Wood [15] showed that code equivalence characterizes finite Frobe-
nius rings. More precisely, MacWilliams [11, 12] proved that linear codes over a
finite field F have the following extension property: if f : C1 → C2 is a linear iso-
morphism between linear codes C1, C2 in Fn
and f preserves the Hamming weight,
then f extends to a monomial transformation of Fn
. It is shown in [15] that if
the extension property holds for linear codes over a finite ring R, then R is a finite
Frobenius ring.
The concept of checkable codes, due to P. Hurley and T. Hurley [7], is related
to the notion zero-divisors in group rings . If RG is a finite group ring, a code
(left ideal) C of RG is said to be checkable if there exists v ∈ RG such that
C = {y ∈ RG : yv = 0}, i.e. if its left annihilator is a principal ideal.
A group ring RG is called code-checkable if every code in RG is checkable.
Jitman et al. [8, Proposition 3.1] showed that if F is a finite field and G is a finite
abelian group, then the group algebra FG is code-checkable if and only if FG is a
2020 Mathematics Subject Classification. Primary 11T71, 16S34, 20C05. Secondary 16L60,
94A24.
Key words and phrases. codes, checkable code, Frobenius ring, quasi-Frobenius ring.
The authors were partially supported by FAPESP, Proc 2015/09162-9.
c
2023 American Mathematical Society
1](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-20-320.jpg)
![2 R. R. DE ARAUJO, A. L. M. PEREIRA, AND C. POLCINO MILIES
principal ideal ring. In [1, Proposition 4.1], Abdelghany and Megahed extended
this result to the case when coefficients are taken in a finite commutative ring.
Recently, Borello, de la Cruz and Willems showed that the same is true for
Frobenius algebras:
[3, Corollary 2.3] In a Frobenius algebra A, a right (resp. left) ideal I is check-
able if and only if Annl(I) resp. Annr(I) is a principal left (resp. right) ideal.
As pointed out in the same paper, there are many examples of checkable codes.
[3, Examples 2.4]
(a) Let e = e2
be an idempotent element in A. Then the ideal eA is checkable.
(b) If A is a semisimple algebra, then all right and left ideals are generated by
idempotents. Thus all right and left ideals are checkable.
(c) All cyclic codes are checkable, since the check equation is given by the check
polynomial.
(d) LCD group codes C (that is, codes for which C ∩C⊥
= {0}) are checkable since
C = eKG with a self-adjoint idempotent e by [4].
The “double annihilator property” plays an important role in [3]. Actually,
this property characterizes finite quasi-Frobenius rings [10]. Following this same
approach, we study checkable codes also over quasi-Frobenius rings.
2. Finite Frobenius and quasi-Frobenius rings
In what follows we consider Frobenius rings and quasi-Frobenius rings only in
the case of finite rings (rings are always associative with identity 1 = 0). A general
definition (including the infinite case) can be found in [10, Chapter 6].
For every left ideal Il R, the right ideal Annr(I) := {x ∈ R : Ix = 0} denotes
the right annihilator of I. Similarly, the left ideal Annl(I) := {x ∈ R : xI = 0} is
the left annihilator of a right ideal I r R.
Definition 2.1. A finite ring R is a quasi-Frobenius ring (QF ring) if it
satisfies the two double annihilator properties:
(i) Annr(Annl(I)) = I, for all right ideal I r R;
(1)
(ii) Annl(Annr(I)) = I, for all left ideal I l R.
(2)
The above definition indicates that, in QF rings, the maps
I r R → Annl(I) l R and I l R → Annr(I) r R define mutually inverse
lattice anti-isomorphisms [10, Corollary 15.6]. Also, denoting by J(R) the Jacob-
son radical of R and by Soc(RR) and Soc(RR) the socles of the R-modules RR and
RR, i.e. the sum of all minimal left and of all minimal right ideals of R respectively,
in [10, Corollary 15.7] it is shown that, in QF rings,
(3) Annl(J(R)) = Soc(RR) = Soc(RR) = Annr(J(R)).
If G is a finite group and R a finite QF ring, then the group ring RG is a QF
ring. A direct sum of rings R =
n
i Ri is a QF if and only if each Ri is a QF
ring. In turn, if F is a finite field, R =
F F
O F
is an example of a ring that is not
quasi-Frobenius [10].
A special class of QF rings are Frobenius rings which, in the finite case, can be
defined as follows:](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-21-320.jpg)
![A NOTE ON CHECKABLE CODES OVER FROBENIUS AND QUASI-FROBENIUS RINGS 3
Definition 2.2. A finite ring R is said to be a Frobenius ring if Soc(RR) is
isomorphic to (R/J(R))R and Soc(RR) is isomorphic to R(R/J(R)).
Theorem 16.14 of [10] guarantees that Frobenius rings are QF rings. Honold,
[6], pointed out important facts about finite Frobenius rings. In particular, in
[6, Theorem 2] it is shown that only one of the isomorphisms in Definition 2.2 is
needed to ensure that a finite ring R is Frobenius:
Theorem 2.1. [6, Theorems 1 and 2] Let R be a finite ring. Then, following
properties are equivalent:
(i) R is Frobenius;
(ii) Soc(RR) is cyclic;
(iii) Soc(RR) is cyclic;
(iv) |I| · |Annr(I)| = |R|, for every left ideal I l R;
(v) |I| · |Annl(I)| = |R|, for every right ideal I r R.
Some examples of Frobenius rings are semisimple rings and finite chain rings.
A direct sum R =
n
i Ri is Frobenius if and only if every Ri is Frobenius. If R is a
Frobenius ring and G is a finite group, then the matrix ring Mn(R) and the group
ring RG are Frobenius rings [10]. Examples of QF ring which are not Frobenius
can be found in [10, Examples 16 and 19].
3. Checkable codes over finite QF rings
Let R be a finite ring. A set C ⊂ R is a left (right) code in R if C is a left
(right) ideal of R. In what follows, we always consider left ideals and hence will
omit the word “left” in the sequel. Similar results hold for right ideals.
The following result gives a characterization of checkable codes in finite QF
rings:
Theorem 3.1. Let R be a finite QF ring. A code C in R is checkable if and
only if Annr(C) is a principal right ideal of R.
Proof. A linear code C in R is checkable if and only if there exists v ∈ R such
that C = Annl(vR). Since R is a QF ring, this is equivalent to the existence of
some element v ∈ R such that Annr(C) = vR, which is a principal right ideal.
Since the annihilator of the socle of a QF ring is its Jacobson radical, we have
the following:
Theorem 3.2. Let R be a finite QF ring. Then:
(i) Soc(RR) is checkable if and only if J(R) is a principal right ideal of R.
(ii) If R is local and J(R) is a principal right ideal of R, then there exists only
one minimal left ideal I of R such that I is checkable.
Proof. The left ideal Soc(RR) is checkable iff there exists v ∈ R such that
Soc(RR) = Annl(vR). Since R is a QF ring, by (3), this is equivalent to vR =
Annr(Soc(RR)) = J(R), which proves (i).
To show (ii), suppose that R is local and J(R) is a principal right ideal of
R. If I is a minimal left ideal of R, then I ⊂ Soc(RR), which implies J(R) =
Annr(Soc(RR)) ⊂ Annr(I). Since R is local, J(R) is maximal and thus J(R) =
Annr(I). So, Annr(I) is principal. By Theorem 3.1, I is checkable. Also, I =](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-22-320.jpg)
![4 R. R. DE ARAUJO, A. L. M. PEREIRA, AND C. POLCINO MILIES
Annl(J(R)) = Soc(RR), which implies that I is the unique minimal left ideal of
R.
4. Code-checkable QF rings
In this section, we characterize code-checkable finite QF rings. Also, we prove
that every code-checkable finite QF ring is a Frobenius ring.
We begin with an elementary result.
Theorem 4.1. Let R be Frobenius ring R of cardinality pn
such that [R : I] = p
and I an ideal such that [R : I] = p, where p is a prime number, n ≥ 2. Then I is
checkable. In particular, if n = 2, the ring R is code-checkable.
Proof. Let I be a left ideal of R such that [R : I] = p. Then |I| = |R|/[R : I] =
pn−1
. Since R is a Frobenius ring, it follows from Theorem 2.1 that |I|·|Annr(I)| =
|R| = pn
. So, |Annr(I)| = p. This means that Annr(I) is a principal right ideal in
R, implying that I is checkable. Now, suppose that n = 2. Let I be a non-trivial
left ideal of R (the cases I = 0 and I = R are trivial). Then |I| divides |R| = p2
and
cannot be 1 or p2
, which means that |I| = p. So, [R : I] = |R|/|I| = p. It follows
from the arguments above that I is checkable. Therefore, R is a code-checkable
ring.
Our next result shows a necessary and sufficient condition for a finite QF ring
to be a Frobenius ring considering checkability of its Jacobson radical.
Theorem 4.2. Let R be a finite QF ring. The Jacobson radical J(R) is check-
able if and only if R is a Frobenius ring.
Proof. If J(R) is checkable, it follows from Theorem 3.1 that Annr(J(R))
is a principal right ideal of R. Since R is a QF ring, it follows from (3) that
Soc(RR) = Annr(J(R)). So, Soc(RR) is a principal right ideal in R, that is,
Soc(RR) is cyclic in the right R-module RR. By theorem 2.1, this implies that
R is Frobenius. Conversely, if R is a Frobenius ring, then R is a QF ring and
Soc(RR) = Annr(J(R)) is cyclic, that is, Annr(J(R)) is a principal right ideal in
R. By Theorem 3.1, J(R) is checkable.
As an immediate consequence of Theorem 4.2 it follows that:
Corollary 4.1. If a finite ring R is a QF ring and code-checkable, then R is
a Frobenius ring.
The next result extends [3, Corollary 2.11] to finite QF rings:
Theorem 4.3. Let R be a finite QF ring. Then every maximal left ideal of R
is checkable.
Proof. For an R-module A, denote by l(A) the composition length of A (that
is, the number of nonzero submodules in a composition series for A). Let M be a
maximal left ideal of R and l = l(R). Then l(M) = l−1. Denote by M∗
the right R-
module HomR(M, R) (the dual module of M) given by the action ϕ(m)r = ϕ(rm),
for all ϕ ∈ M∗
, r ∈ R and m ∈ M. It follows from duality properties (e.g., see
[10, Equation 15.14]) that R/Annr(M) is isomorphic to M∗
. Since l(M∗
) = l(M),
then l(Annr(M)) = 1, which means that Annr(M) is a minimal right ideal of R.
Therefore, Annr(M) is a right principal ideal of R, since every minimal ideal is
principal. It follows from Theorem 3.1 that M is checkable.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-23-320.jpg)
![A NOTE ON CHECKABLE CODES OVER FROBENIUS AND QUASI-FROBENIUS RINGS 5
The following theorem shows that, for QF rings, the class of code-checkable
rings is precisely the class of principal right ideal rings. In particular, this means
that a commutative QF ring is code-checkable iff it is a principal ideal ring extending
[3, Theorem 3.1] and [1, Theorem 4.5].
Theorem 4.4. Let R be a finite QF ring. Then, R is code-checkable if and
only if R is a principal right ideal ring.
Proof. Suppose that R is a code-checkable ring. Let I be a right ideal in
R. Since Annl(I) is a linear code (left ideal) and R is a QF ring, Theorem 3.1
implies that I = Annr(Annl(I)) is principal. Therefore, R is a principal right ideal
ring. Conversely, let C be a linear code of R. As R is a principal right ideal ring,
Annr(C) is principal. It follows from Theorem 3.1 that C is checkable. Therefore,
R is code-checkable.
Corollary 4.2. Consider the direct sum R =
n
i=1 Ri of finite commutative
QF rings. Then, R is code-checkable if and only if Ri is code-checkable, for all
i, 1 ≤ i ≤ n.
Proof. Since each Ri is a QF ring, for all 1 ≤ i ≤ n, R is also a finite QF
ring. From Theorem 4.4, R is a code-checkable ring iff R is a principal ideal ring.
Also, since R is commutative, R is a principal ideal ring iff every Ri is a principal
ideal ring, that is, every Ri is code-checkable and the result follows.
Corollary 4.3. Let R be a finite, commutative and code-checkable QF ring.
Let C be a code in R. If C = Annr(C) (= Annl(C)), then C is a nilpotent ideal
of R.
Proof. By Theorem 4.4, R is a principal ideal ring. Then, for each code C
in R, there exists v ∈ R such that Annr(C) = vR = Rv, which implies Cv = 0. If
C = Annr(C), then C = Rv and v2
= 0. Therefore, C is nilpotent.
References
[1] Noha Abdelghany and Nefertiti Megahed, Code-checkable group rings, J. Algebra Comb. Dis-
crete Struct. Appl. 4 (2017), no. 2, 115–122, DOI 10.13069/jacodesmath.284939. MR3601343
[2] S.D. Berman, On the theory of group codes, Kibernetika,1 (1967), 31-39.
[3] Martino Borello, Javier De La Cruz, and Wolfgang Willems, On checkable codes in
group algebras, J. Algebra Appl. 21 (2022), no. 6, Paper No. 2250125, 11, DOI
10.1142/S0219498822501250. MR4429010
[4] Javier de la Cruz and Wolfgang Willems, On group codes with complementary duals, Des.
Codes Cryptogr. 86 (2018), no. 9, 2065–2073, DOI 10.1007/s10623-017-0437-2. MR3816214
[5] Marcus Greferath, Alexandr Nechaev, and Robert Wisbauer, Finite quasi-Frobenius modules
and linear codes, J. Algebra Appl. 3 (2004), no. 3, 247–272, DOI 10.1142/S0219498804000873.
MR2096449
[6] Thomas Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001),
no. 6, 406–415, DOI 10.1007/PL00000451. MR1831096
[7] P. Hurley and T. Hurley, Module codes in group rings. Proc. Int. Symp. Information Theory
(ISIT), 2007, pp. 1981-1985.
[8] S. Jitman; S. Ling; H. Liu; X. Xie. Checkable Codes from Group Rings, arXiv:1012.5498,
2010.
[9] Michael Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4 (German),
Arch. Math. (Basel) 53 (1989), no. 2, 201–207, DOI 10.1007/BF01198572. MR1004279
[10] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189,
Springer-Verlag, New York, 1999, DOI 10.1007/978-1-4612-0525-8. MR1653294](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-24-320.jpg)
![6 R. R. DE ARAUJO, A. L. M. PEREIRA, AND C. POLCINO MILIES
[11] Jessie MacWilliams, Error-correcting codes for multiple-level transmission, Bell System Tech.
J. 40 (1961), 281–308, DOI 10.1002/j.1538-7305.1961.tb03986.x. MR141541
[12] Florence Jessie MacWilliams, Combinatorial Problems of Elementary Abelian Groups, Pro-
Quest LLC, Ann Arbor, MI, 1962. Thesis (Ph.D.)–Radcliffe College. MR2939359
[13] F. J. MacWilliams, Binary codes which are ideals in the group algebra of an Abelian
group, Bell System Tech. J. 49 (1970), 987–1011, DOI 10.1002/j.1538-7305.1970.tb01812.x.
MR265054
[14] E. Prange, Cyclic error-correcting codes in two symbols, AFCRC-TN-57-103, USAF, Cam-
bridge Research Laboratories, New York (1957)
[15] Jay A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc.
136 (2008), no. 2, 699–706, DOI 10.1090/S0002-9939-07-09164-2. MR2358511
Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, CEP 11533-160,
Brazil.
Email address: robson.ricardo@ifsp.edu.br
Demat, Universidade Federal Rural do Rio de Janeiro, Soropedica, RJ, Brazil, CEP
Seropédica - RJ, 23890-000
Email address: almp1980@ufrrj.br
Instituto de Matemática e Estatı́stica, Universidade de São Paulo, Caixa Postal
66281, CEP-05315-970, Brazil.
Email address: polcinomilies@gmail.com](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-25-320.jpg)
![Contemporary Mathematics
Volume 785, 2023
https://doi.org/10.1090/conm/785/15771
Good classical and quantum codes from multi-twisted codes
Nuh Aydin, Thomas Guidotti, and Peihan Liu
Dedicated to Professor Syed Tariq Rizvi for his contribution to Ring and Module Theory
Abstract. Multi-twisted (MT) codes were introduced as a generalization of
quasi-twisted (QT) codes. QT codes have been known to contain many good
codes. In this work, we show that codes with good parameters and desir-
able properties can be obtained from MT codes. These include best known
and optimal classical codes with additional properties such as reversibility
and self-duality, and new and best known non-binary quantum codes obtained
from special cases of MT codes. Often times best known quantum codes in
the literature are obtained indirectly by considering extension rings. Our con-
structions have the advantage that we obtain these codes by more direct and
simpler methods. Additionally, we found theoretical results about binomials
over finite fields that are useful in our search.
1. Introduction and Motivation
A liner code C of length n over Fq, the finite field of order q (also denoted by
GF(q)), is a vector subspace of Fn
q . The three important parameters of a linear code
are: the length (n), the dimension (k), and the minimum distance (d). If a code C
over Fq is of length n, dimension k, and minimum distance d, then we refer to it as
an [n, k, d]q-code. One of the most important and challenging problems in coding
theory is to determine the optimal parameters of a linear code and to explicitly
construct codes that attain them. For example, given the alphabet size q, length
n, and dimension k, researchers try to determine the highest value of the minimum
distance d, and also to give an explicit construction of such a code. Despite much
work on this question, it still has many open cases. There are databases that contain
information about best known linear codes (BKLC) and their constructions. Two
of the well known databases are the maintained by M. Grassl ( [21], codetables.de)
and the database contained in the Magma software [25].
Given q, n and k, there are theoretical upper bounds on d that are given in
the tables mentioned above. However, it is not guaranteed and it is not always
possible that they can actually be attained. This is a challenging problem for
two main reasons: firstly, computing the minimum distance of a linear code is an
2020 Mathematics Subject Classification. Primary 94B05, 94B60.
Key words and phrases. Multi-twisted Codes, best known linear codes, quantum codes, re-
versible codes, self-dual codes.
The authors were supported in part by a Kenyon Summer Science Scholars program.
c
2023 American Mathematical Society
7](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-26-320.jpg)
![8 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU
NP-hard problem [32] and secondly, the number
(qn
− 1)(qn
− q) · · · (qn
− qk−1
)
(qk − 1)(qk − q) · · · (qk − qk−1)
of linear codes over Fq of length n and dimension k grows fast when n, k and q
increase. Therefore, it is impossible to do an exhaustive search except for very small
parameters. Hence, researchers focus on sub-classes of linear codes with special
algebraic properties. One very promising class has been the class of quasi-twisted
(QT) codes which contains the important classes such as cyclic codes, constacyclic
codes, and quasi-cyclic (QC) codes as special cases. Hundreds of BKLCs have been
obtained, usually by computer searches, from the class of QT codes.
More recently, a generalization of QT codes called multi-twisted (MT) codes
was introduced in [4]. Therefore, MT codes are a generalization of cyclic, negacyclic,
constacyclic (CC), quasi-cyclic (QC), generalized quasi-cyclic (GQC) and quasi-
twisted codes (QT). Compared to QT codes, the search space for MT codes is
much larger. While this increases the chances of finding new codes, it also makes
the search process computationally more expensive. In this work, we focus on a
special type of MT codes whose generator matrix is of the form G = [Ik | A] where
Ik is the k × k identity matrix and A is a k × (n − k) circulant matrix. The idea of
investigating these codes was first introduced in [10]. In addition to trying to find
new linear codes from this type of MT codes, we also worked on obtaining codes
that are reversible, self-dual, or self-orthogonal.
Finally, we also worked on obtaining quantum codes from classical codes, specif-
ically from constacyclic, QC, QT, and MT codes. The idea of quantum codes was
first introduced in [28]. Later, methods of constructing quantum codes from clas-
sical codes were introduced in [15] (known as the CSS construction) and [30] (the
Steane construction). Since then, researchers have been working on various ways of
obtaining quantum codes from classical codes. More recently, codes over extension
rings have attracted a lot of attention. Many quantum codes in the literature have
been obtained via the method of considering codes over an extension ring S of a
base/ground ring (or field) R and using a map to eventually obtain codes over the
ground ring R. We showed that in many cases, there is no need to obtain these
codes in such an indirect way. We have obtained many of these best known codes
presented in the literature, and even better ones, by using a direct approach. It is
preferable to obtain codes in more direct ways with convenient algebraic structures.
2. Preliminaries
Definition 2.1. [4] For each i = 1,. . . , , let mi be a positive integer and
ai ∈ F∗
q = Fq {0}. A multi-twisted (MT) module V is an Fq[x]-module of the form
V =
i=1
Fq[x]/xmi
− ai.
A MT code is an Fq[x]-submodule of a MT module V .
Equivalently, we can define a MT code in terms of the shift of a codeword.
Namely, a linear code C is MT if for any codeword
c = (c1,0, . . . , c1,m1−1; c2,0, . . . , c2,m2−1; . . . ; c,0, . . . , c,m−1) ∈ C,
its multi-twisted shift
(a1c1,m1−1, c1,0, . . . , c1,m1−2; a2c2,m2−1, c2,0, . . . , c2,m2−2; . . . ; ac,m−1, . . . , c,m−2)](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-27-320.jpg)
![GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 9
is also a codeword. If we identify a vector
c with C(x) = (c1(x), c2(x), . . . , c(x))
where ci(x) = ci,0 + ci,1x + · · · + ci,mi−1xmi−1
, then the MT shift corresponds to
xC(x) = (xc1(x) mod xm1
− a1, . . . , xc(x) mod xm
− a).
The following are some of the most important special cases of MT codes:
• a1 = a2 = · · · = a gives QT codes
• a1 = a2 = · · · = a = 1 gives QC codes
• = 1, a1 = 1 gives cyclic codes
• = 1, a1 = −1 gives negacyclic codes
• = 1 gives constacyclic codes
There has been much research on QT codes, a special case of MT codes. Hun-
dreds of new codes have been obtained from QT codes by computer searches. An
algorithm called ASR ([3]) has been particularly effective in this regard. The ASR
algorithm is based on the following theorem.
Theorem 2.2. [3] Let C be a 1-generator QT code of length n = m over Fq
with a generator of the form
(g(x)f1(x), g(x)f2(x), ..., g(x)f(x))
where xm
− a = g(x)h(x) and gcd(h(x), fi(x)) = 1 for all i = 1, . . . , . Then
dim(C) = m − deg(g(x)), and d(C) ≥ · d where d is the minimum distance of the
constacyclic code Cg generated by g(x).
This algorithm has been refined and automatized in more recent works such as
([5], [6], [7], [8], [9]) and dozens of record breaking codes have been obtained over
every finite field Fq, for q = 2, 3, 4, 5, 7, 8, 9 through its implementation. Moreover,
it has been further generalized in [10] and more new codes were discovered that
would have been missed by its earlier versions.
We can consider MT analogues of the QT codes given in the previous theorem.
The following theorem gives a basic fact about duals of such codes.
Theorem 2.3. Let C = (g1(x), g2(x), . . . , g(x)) be an MT code such that
the constacyclic codes generated by gi(x) have the same length n and the same
dimension k for i = 1, 2, . . . , . Let Di be the dual of gi(x). Let PD = D1 ×
D2 × · · · × D. Then PD is contained in C⊥
.
Proof. Let Gi be a generator matrix of Ci = gi(x), so a generator matrix
G of C is given by
G =
G1 G2 G3 . . . G
k×
i=1 ni
.
Let Hi be a generator matrix of Di = C⊥
i . Then a generator matrix GPD of PD
is given by
GPD =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
H1 0 0 . . . 0
0 H2 0 . . . 0
0 0 H3 . . . 0
...
...
...
...
...
0 0 0 . . . H
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(n−k)×
i=1 ni
.
It is readily verified that G·GPD = 0. Note that the dimension of C⊥
is
i=1 ni−k
and the dimension of PD is
i=1(ni − k), so the dimension of PD is less than or
equal to the dimension of C⊥
. Hence, PD ⊆ C⊥
.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-28-320.jpg)
![10 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU
3. A Special Type of MT Codes
We first introduced a special type of MT codes that we called ICY codes in
[10]. The basic idea for this construction is inspired by the well known fact that
any linear code is equivalent to a linear code whose generator matrix is of the
form G = [Ik | A] where Ik is the k × k identity matrix and A is a k × (n − k)
matrix. A code with a generator matrix in this form is called a systematic code.
In this search method, we begin with a k × k identity matrix on the left and
append it with k × (n − k) circulant matrices on the right. The circulant matrix
is constructed using a generator polynomial g(x)|(xn−k
− a) of a constacyclic code
(obtained using the partition program from [9]). For each polynomial g(x), we
consider a number of circulant matrices obtained from polynomials g(x)·f(x) where
gcd(f(x), h(x)) = 1, with the goal of finding a code with minimum distance equal to
or greater than the BKLCs for parameters [n, k]q. In [10] we were able to find many
good codes over GF(2) and GF(5). In this paper we present a number of optimal
codes over GF(2) with parameters [2·k, k] obtained from an implementation of this
method. A number of these codes we found are also self-dual, which makes them
even more useful. Another property of a code that is useful in some applications,
such as DNA codes is reversibility, is reversibility. A code is reversible if, whenever
(c0, c1, . . . , cn−1) is a codeword, so is (cn−1, cn−2, . . . , c0).
Lemma 3.1. If f(x) is a monomial and the block length of f(x) is the same as
the block length of the identituy matrix, then the MT code C over GF(2) generated
by 1, f(x) is always self-dual, reversible and of minimum distance 2. In particular,
if the monomial is a constant, then C is a cyclic code.
Proof. Let f(x) = xi
be a monomial and the block length of f(x) be the
same as k × k identity matrix. Then the generator matrix of the MT code C over
GF(2) generated by 1, f(x) is given by:
G =
⎛
⎜
⎜
⎜
⎝
1 0 . . . 0 0 . . . 1 0 . . .
0 1 . . . 0 0 . . . 0 1 . . .
...
...
...
...
...
...
...
...
...
0 0 . . . 1 . . . 1 0 0 . . .
⎞
⎟
⎟
⎟
⎠
k×2k
.
It is easy to verify that the reverse of the jt
h row is the (k − i + 1 − j)t
h row,
so it follows that C is reversible. Also note that each row of G is orthogonal to
itself and also orthogonal to every other row, so given k = 2k/2 we know C is also
self-dual. Moreover, since each row is of weight 2, and both matrices are circulant,
it follows that C is of minimum distance 2.
Specifically, if f(x) = 1, then the generator matrix G is given by
G =
⎛
⎜
⎜
⎜
⎝
1 0 . . . 0 1 . . . 0 . . .
0 1 . . . 0 0 1 0 . . .
...
...
...
...
...
...
...
...
0 0 . . . 1 . . . 0 . . . 1
⎞
⎟
⎟
⎟
⎠
k×2k
.
Hence, the code is a cyclic code of length 2k generated by
f(x) = (xk
− 1)|(x2k
− 1).
The following codes in Table 1 are obtained by the ICY method where we chose
n = 2k, so each one of these codes has parameters [2k, k, d]. All of these codes are](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-29-320.jpg)
![GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 11
optimal with additional properties of being reversible and/or self orthogonal. For
the circulant matrix on the right, we used a random m × m circulant matrix by
choosing a random vector and taking its cyclic shifts. This polynomial is given in
the table. We also checked the conditions for reversibility and self-duality. Note
that self-orthogonality is the same as self-duality since we set n = 2k. In this
search, the binary field fields turned out to be most promising.
In this and every subsequent table presented in this paper the terms of the
generator polynomials are listed in ascending order from left to right. For instance,
the generator polynomial [111101] of the fourth code in this table is the vector
representation of g(x) = 1 + x + x2
+ x3
+ x5
.
Table 1: Optimal codes from the ICY method with several desirable
properties. Here, ∗
delineates a reversible code, and ◦
delineates a self-
dual code.
[n, k, d]q Polynomials
[8, 4, 4]2
∗◦
g = [111]
[10, 5, 4]2 g = [10101]
[12, 6, 4]2
∗◦
g = [110111]
[14, 7, 4]2 g = [111101]
[20, 10, 6]2 g = [11111011]
[24, 12, 8]2
∗◦
g = [101111011]
[28, 14, 8]2 g = [11101111001001]
[32, 16, 8]2
∗◦
g = [1100100001100011]
[36, 18, 8]2
∗◦
g = [111010010111001]
[42, 21, 10]2 g = [11110011011011101001]
[48, 24, 12]2
∗◦
g = [10111101001011110011101]
[16, 8, 6]3
∗
g = [212222]
[18, 9, 6]3
∗
g = [201002222]
[20, 10, 7]3
∗
g = [112012102]
[30, 15, 9]3
∗
g = [201212222200121]
[14, 7, 6]4
∗◦
g = [ααα2
αα2
α2
]
[20, 10, 8]4
∗◦
g = [αα1α00α2
1α2
α2
]
[24, 12, 9]4 g = [10αα2
αα2
α10α1]
[14, 7, 6]5 g = [3343442]
[16, 8, 7]5 g = [34404411]
[12, 6, 6]7 g = [551641]
[14, 7, 7]7 g = [5226211]
[16, 8, 7]7 g = [42325344]
[18, 9, 8]7 g = [524635101]
The next table presents a list of optimal self-dual codes together with their
weight enumerators obtained by this method. For codes over F4 = {0, 1, α, α + 1}
in this paper, α is a root of x2
+x+1 over F2. Again, ∗
delineates a reversible code.
The table also presents the weight enumerator of each code and for each binary
code specifies whether it is Type I or Type II.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-30-320.jpg)
![12 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU
Table 2: Optimal self-dual codes from the ICY method.
[n, k, d]q Type weight enumerator Polynomials
[8, 4, 4]2
∗ II A8
+ 14A4
+ 1 g = [111]
[12, 6, 4]2 I A12
+ 15A8
+ 32A6
+ 15A4
+ 1 g = [110111]
[16, 8, 4]2
∗ II A16
+ 28A12
+ 198A8
+ 28A4
+ 1 g = [110111]
[16, 8, 4]2
∗ I A16
+12A12
+64A10
+102A8
+64A6
+
12A4
+ 1
g = [1010111]
[18, 9, 4]2
∗ I A18 + 9A14 + 75A12 + 171A10 +
171A8 + 75A6 + 9A4 + 1
g = [11011001]
[20, 10, 4]2
∗ I A20
+ 45A16
+ 210A12
+ 512A10
+
210A8
+ 45A4
+ 1
g = [1011111111]
[22, 11, 6]2
∗ I A22
+ 77A16
+ 330A14
+ 616A12
+
616A10
+ 330A8
+ 77A6
+ 1
g = [1001011]
[26, 13, 6]2 I A26
+ 52A20
+ 390A18
+ 1313A16
+
2340A14
+ 2340A12
+ 1313A10
+
390A8
+ 52A6
+ 1
g = [10111110111]
[52, 26, 10]2 I A52
+442A42
+6188A40
+53040A38
+
308958A36
+ 1270360A34
+
3754569A32
+ 8065616A30
+
12707500A28
+ 14775516A26
+
12707500A24
+ 8065616A22
+
3754569A20
+ 1270360A18
+
308958A16
+ 53040A14
+ 6188A12
+
442A10
+ 1
g =
[1001111100101010100100101]
[14, 7, 6]4
∗ Euclidean 318A14
+ 1302A13
+ 2940A12
+
3990A11
+ 3738A10
+ 2226A9
+
1155A8
+ 546A7
+ 168A6
+ 1
g = [α1αα2ααα2]
[16, 8, 6]4
∗ Euclidean 681A16
+ 3696A15
+ 7896A14
+
14112A13
+ 16464A12
+ 9408A11
+
7392A10
+4704A9
+678A8
+336A7
+
168A6
+ 1
g = [αα1α2αα2]
[20, 10, 8]4
∗ Euclidean 171A12
+432A11
+864A10
+1440A9
+
459A8
+ 432A7
+ 288A6
+ 9A4
+ 1
g = [αα1α00α2α2α2]
[22, 11, 8]4
∗ Euclidean 171A12
+432A11
+864A10
+1440A9
+
459A8
+ 432A7
+ 288A6
+ 9A4
+ 1
g = [α2ααα2010αα2α2α]
4. Reversible Codes
Reversible codes are essential in application of coding theory to DNA computing
[27]. Each single DNA strand is composed of a sequence of four bases nucleotides
(adenine (A), guanine (G), thymine (T), cytosine (C) ) and it is paired up with
a complementary strand to form a double helix [1]. Finding reversible codes is
an essential requirement in order to find codes suitable for DNA computing. Also,
reversible codes may be used in certain data storage applications [26]. For example,
if the code is reversible then one can read the stored data from either end of a block.
In this paper, we present a general method to construct reversible -block MT, QT
and QC codes, and a number of good reversible QC codes over GF(2) are obtained
using this method.
The codes presented in Table 3 are reversible and self-orthogonal QC with
the same parameters as BKLCs over GF(2). Hence, they are more desirable than
BKLCs that do not have these properties. In checking the reversibility of these
codes, we used the conditions for reversibility of QC codes given in [31]. Theo-
rem 1 in section IV of [31] presents three conditions for reversibility of QC codes
with generator matrices of the form [g(x), g(x) · f(x)], where g(x)|(xm
− 1) and
gcd(f(x), h(x))) = 1. After finding all nonequivalent generators g(x) for a given](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-31-320.jpg)
![GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 13
block length m, we check the first two conditions on g(x) and h(x) before proceed-
ing. If these two checks pass, then we generate random polynomials f(x) that satisfy
the third condition in the theorem, resulting in a reversible QC code. Additionally,
since these QC codes are constructed over GF(2) they have the added property of
being self-orthogonal. Theorem 1 also provides conditions for self-orthogonality of
QC codes with such generator matrices and in the binary case these conditions are
the same as those for reversibility. For a polynomial f(x), it is customary to denote
its reciprocal by f(x)∗
.
Theorem 4.1. Let gi|(xni
− ai) and gcd(hi, fi) = 1 for i = 1, 2, . . . , ,
where hi = xni −ai
gi
. If deg(gifi) = deg(g+1−if+1−i), ni = n+1−i and gifi =
(g+1−if+1−i)∗
for i = 1, 2, .., , then the MT code C generated by (g1f1, . . . , gf)
is reversible.
Proof. Let gifi = ai,0 + ai,1x + · · · + ai,ki
xki
, so a generator matrix of this
MT code with blocks is given by
G =
G1 G2 . . . G−1 G
,
where Gi is
Gi =
⎛
⎜
⎜
⎜
⎝
ai,0 ai,1 ai,2 . . . ai,ki
0 0 . . .
0 ai,0 ai,1 . . . ai,k−1 ai,ki
0 . . .
.
.
.
...
...
...
...
...
...
...
0 0 . . . ai,0 ai,1 ai,2 . . . ai,ki
⎞
⎟
⎟
⎟
⎠
(ni−ki)×ni
.
Hence, the reversed first row of G is given by
(0, . . . , 0, a,k
, a,k−1, . . . , a,1, a,0;
0, . . . , 0, a−1,k−1
, a−1,k−1−1, . . . , a−1,1, a−1,0;
.
.
.
0, . . . , 0, a1,k1
, a1,k1−1, . . . , a1,1, a1,0).
Note that gifi = (g+1−if+1−i)∗
for i = 1, 2, .., , so ai,j = a+1−i,k−j+1 for
i = 1, 2, .., and j = 1, 2, . . . , k. Also, since the degree of all deg(gifi) =
deg(g+1−if+1−i), it is not hard to see that the reversed first row is the (n − k)th
row, and in general the reversed jth
row is the same as (n − k + 1 − j)th
row.
Therefore, the reverse of each row in G is in G, so C is reversible.
As mentioned previously all of these codes are constructed using the conditions
for a 2-QC code to be reversible found in [31]. This paper gives conditions on the
generator polynomials g(x) and f(x) · g(x) mod (xm
− 1) with gcd(h(x), f(x)) = 1
so the first polynomial presented in this table is g(x), and the second one is f(x)·g(x)
mod (xm
− 1).](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-32-320.jpg)
![14 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU
Table 3: Reversible and self-orthogonal binary QC codes with best
known parameters
[n, k, d] Polynomials
[60, 28, 12] [00101100111001101100101001101], [111]
[60, 29, 12] [000111000101011000100100011111], [11]
[72, 29, 16] [011111101110101111101101100111101011], [10111101]
[72, 30, 16] [0110111100000111111100010010111001], [1101011]
[80, 37, 16] [1010011110001011110000000111110001110001], [1111]
[80, 38, 16] [0011111110101010100100101101110101011], [101]
[84, 24, 24] [111100010111100111000111101110000000100101],
[1010100000000010101]
[84, 25, 24] [001111100001011100110011110100001101000111],
[110011111111110011]
[84, 26, 24] [001101010011000011101100000100001110001101],
[11100000000000111]
[84, 27, 24] [001001101000110101101001110110000011000001],
[1011111111111101]
[90, 28, 24] [011111010111100100001011100011000011110000001],
[110111011110111011]
[90, 34, 20] [00010111011111110110000000011010111111101101],
[110001100011]
[96, 36, 20] [10010100100000001100000001110111001001100111001],
[1010001000101]
[96, 37, 20] [01100110011010101011011110010011101110010010011],
[100110011001]
[96, 38, 20] [110011110011101110010010111100100111011010101],
[11101110111]
[100, 41, 20] [00110100011011100110001010100100101010000111001001],
[1111111111]
[102, 34, 24] [10011111000011100010010001100001100001101110101],
[101110111111011101]
[102, 42, 20] [1010110010011100000100111101101100110001000000101],
[1001111001]
[104, 36, 24] [1010111000101110001001010010110111001001101110110101],
[11110000000001111]
[108, 46, 20] [110000001010000001101111000101010010011011101000001],
[111111111]
[110, 40, 24] [110111111111101000011111110110011000010001011111000111],
[1111100000011111]
5. Quantum Codes
In comparison to classical information theory, the field of quantum information
theory is relatively young. The idea of quantum error correcting codes was first
introduced in [29] and [14]. A method of constructing quantum error correcting
codes (QECC) was given in [15]. Since then researchers have investigated various
methods of using classical error correcting codes to construct new QECCs. The
majority of the methods have been based on the CSS construction given in [15]. In
this method, self-dual, self-orthogonal and dual-containing linear codes are used to
construct quantum codes. The CSS construction requires two linear codes C1 and
C2 such that C⊥
2 ⊆ C1. Hence, if C1 is a self-dual code, then we can construct a](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-33-320.jpg)
![GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 15
CSS quantum code using C1 alone since C⊥
1 ⊆ C1. If C1 is self-orthogonal, then
we can construct a CSS quantum code with C⊥
1 and C1 since C1 ⊆ C⊥
1 . Similarly
in the case C1 is a dual-containing code. The last few tables in this section present
a number of good QECCs obtained from both the CSS method and the method of
additive codes. Many of these codes have the same parameters as those found in the
literature, however we obtained them using more direct and simpler constructions.
Additionally, we have found other codes whose parameters do not appear in the
literature.
The following codes are constructed using the method of quantum codes con-
structed from additive codes [15]. After generating all divisors g(x) of
xm
− a that generate non-equivalent constacyclic codes and finding a generator
matrix of the form [Cg(x)·f1(x), Cg(x)·f2(x)] where gcd(fi(x), h(x))) = 1 and Cg(x)
represents the circulant matrix from g(x), we construct an additive code from this
generator matrix and test it for symplectic self orthogonality. If the code passes
this check, then we construct a quantum code from this additive code and compute
its minimum distance. Then we can compare the minimum distances of these codes
against the comparable best known QECCs given in the database codetables.de.
In Table 4, all of the codes are optimal quantum QT codes constructed over
GF(22
). Note that α is a root of the irreducible polynomial x2
+x+1 over GF(2).
Table 4 : Optimal quantum codes over GF(22
) from QT Codes
[n, k, d] Polynomials
[[54, 47, 2]] [00α2
1110α000α2
1110α000α2
1110α],
[α2
αα2
01α2
α0αα2
αα2
01α2
α0αα2
αα2
01α2
α0α]
[[56, 50, 2]] [10000α2
αα1α2
1α2
α1α2
αααα100α2
1α2
10α2
],
[α2
10011ααα2
1α0α2
α2
0αα2
α2
αα110α1α2
]
[[70, 64, 2]] [10α2
α010α0αα1ααα001ααα2
10α0α1α2
00α2
110α],
[α2
01α2
ααα2
αα2
101α2
α2
01α2
α2
0α2
αα1011α0α2
α2
α2
1α0α]
[[88, 80, 2]] [100α2
1αα101α2
ααα2
1α01α2
α2
α2
0011α0α2
α2
010αα2
α2
α0α2
10ααα1],
[0α2
11α00α2
000100αα2
α2
101α2
1α1α2
α2
0αα1αααα2
αα011α2
αα2
1α2
]
[[98, 91, 2]] [11α2
101011α2
101011α2
101011α2
101011α2
101011α2
101011α2
101],
[αα000α2
0αα000α2
0αα000α2
0αα000α2
0αα000α2
0αα000α2
0αα000α2
]
We also found some QECCs whose parameters do not appear in the literature.
Hence, we consider them new quantum codes. The inspiration for this idea came
from [17]. These codes are obtained using direct sums of two cyclic codes, with
generators g1(x), g2(x) that are divisors of xm
− 1. Each gi(x) generates a dual
containing cyclic code Ci = gi(x) which implies that their direct sum C1 × C2 is
also dual containing. So, the CSS method can be used with the direct sum code and
its dual. The resultant QECC will have parameters [[2 · m, 2 · k − 2 · m, d]]q. Here,
k is defined to be the dimension of the direct sum code which is equal to k1 + k2,
the sum of the dimensions of the individual cyclic codes, and d is the minimum of
the minimum distances of the cyclic codes generated by g1(x) and g2(x). Table 5
contains these new codes.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-34-320.jpg)
![16 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU
Table 5 : New QECCs
[[n, k, d]]q Polynomials
[[60, 54, 2]]3 [21], [201]
[[72, 68, 2]]3 [21], [21]
[[84, 80, 2]]3 [21], [21]
[[100, 96, 2]]5 [41], [41]
Next, we have a number of codes that have the same parameters as the best
known QECCs presented in the literature. Our codes have the advantage that they
are obtained via a more direct and simpler construction. They were all obtained
using the same method as the codes in the previous table. Codes with the same
parameters listed in Table 6 were found using more complex and indirect methods
detailed in [11], [2], [24] , [18], [19], [13]. These constructions usually involve
considering an extension ring S of a ground ring or field R, constructing a code
over the extension ring, then coming back to R using some sort of Gray map. We
argue that it is more desirable to construct codes with the same parameters using
a more direct approach.
Table 6 : QECC ties
[[n, k, d]]q Polynomials Source
[[72, 66, 2]]3 [21], [111] [24]
[[84, 78, 2]]3 [21], [201] [11]
[[40, 36, 2]5 [41], [41] [19]
[[40, 34, 2]]5 [41], [131] [11]
[[40, 28, 3]]5 [3011], [3011] [19]
[[60, 54, 2]]5 [41], [131] [24]
[[90, 84, 2]]5 [41], [131] [18]
[[100, 94, 2]]5 [41], [401] [11]
[[140, 134, 2]]5 [41], [401] [13]
5.1. New Quantum Codes from Constacyclic Codes and a Result
About Binomials. In this section, we obtain new QECCs that are constructed
by the method of CSS construction from constacyclic codes over finite fields. These
codes are new in the sense that either there do not exist codes with these param-
eters in the literature (to the best of our knowledge) or the minimum distances of
our codes are higher than the codes that are presented in the literature. Our work
on this also led to a couple of theoretical results about the binomials of the form
xn
− a over Fq. The first one (Lemma 2 below) is similar to Theorem 4.1 in [4].
We start with the results on binomials.
We will be using the fact that the Frobenius map x → xpm
on a finite field Fq of
characteristic p is a permutation of Fq for any positive integer m. In particular, for
any α ∈ Fq and for any m ∈ Z+
, there exists a unique β ∈ Fq such that α = βpm
.
Lemma 5.1. Let n ∈ Z+
, q be a prime power and a, b ∈ Fq such that a = b.
Then gcd(xn
− a, xn
− b) = 1.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-35-320.jpg)
![GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 17
Proof. It suffices to show that the polynomials xn
− a and xn
− b have no
common roots. Let δ be a root of xn
− a. Then,
δn
− b = a − b = 0
It follows that gcd(xn
− a, xn
− b) = 1.
Theorem 5.2. Let q be a power of prime p, and let g(x) = xn
− a ∈ Fq[x].
Then all irreducible factors of xn
− a have the same multiplicity pz
n, where z is
the largest positive integer such that pz
divides n.
Proof. Let n = pz
· b, where gcd(b, p) = 1. Then xn
− a can be written as
xn
− a = xpz
·b
− a
= (xb
− a
)pz
,
where a
∈ Fq. Let’s call f(x) = xb
− a
, and then consider gcd(f(x), f
(x)). Note
that b = 0 and xb
− a
is not a multiple of x, so it follows that
gcd(f(x), f
(x)) = gcd(xb
− a
, bxb−1
) = 1. Hence, all irreducible factors of f(x) =
xb
− a
have the same multiplicity 1. Therefore, all irreducible factors of g(x) =
xn
− a = fpz
(x) have the same multiplicity pz
.
By the definition of CSS construction, we need two codes such that one is
contained in the dual of the other one. By the well known fact about inclusion of
ideals, we know that
g(x)f(x) ⊆ g(x)
and we consider C⊥
2 = g(x)f(x) ⊆ g(x) = C1. Here g(x) is a divisor of a
binomial xn
− a and it generates a constacyclic code.
Since for a given n ∈ Z+
and distinct elements α, β ∈ Fq, gcd(xn
−α, xn
−β) =
1, it follows that for any divisor g(x)|(xn
− α), we have g(x)f(x) (xn
− β) for
any f(x) ∈ Fq[x]. In other words, for a given C1 = g(x), where g(x)|(xn
− α),
all possible constacyclic codes of the same length C⊥
2 such that C⊥
2 ⊆ C1 have the
form C⊥
2 = g(x)f(x), where g(x)f(x)|(xn
− α). Hence, for given α ∈ Fq and n,
we only need to consider gi(x)|(xn
− α) and gi(x)fi(x)|(xn
− α). By Lemma 5.1,
any generator of C⊥
2 will be a divisor of xn
− α, but not of xn
− β for β = α. Also,
based on the multiplicity of all irreducible factors of xn
− α, for any g(x), we can
determine all g(x)f(x) by enumerating all possible combinations of multiplicities
of irreducible factors. Therefore, our algorithm goes through every possible pair of
g(x) and g(x)f(x), and we view each g(x) as C1 and each g(x)f(x) as C⊥
2 in
the CSS construction. This way, we obtained many new QECCs. The new codes
we obtained are of two types: i) those that have higher minimum distances than
the codes given in the literature, ii) those whose parameters do not appear in the
literature. We present them in the following two tables. Table 7 contains references
to the literature where a comparable code is given.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-36-320.jpg)
![18 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU
Table 7 : New QECCs that have higher minimum distances than the
codes given in the literature
KnownQECC Ref New
[[n, k, d]]q
g(x) g(x)f(x)
[[11, 1, 4]]5 [23] [[11, 1, 5]]5 [124114], [1122031]
[[30, 20, 3]]5 [24] [[30, 20, 4]]5 [142241], [14401243413033031434210441]
[[40, 32, 2]]5 [2] [[40, 32, 3]]5
[12342], [12030222342024440142411123141333
40313]
[[93, 75, 2]]5 [12] [[93, 75, 4]]5
[1032243334],[1214314012224022200134214311
4334244231242301201241431231121012330223
41423121100111211]
[[60, 52, 2]]5 [24] [[60, 52, 3]]5
[10434],[101321202243220133133431344440014
124413202
440312032014321]
[[78, 60, 3]]3 [12] [[78, 60, 4]]5
[1120201112],[12022010222111011212012012102
20211120012212110210210000100012101020022]
[[18, 2, 3]]7 [22] [[18, 2, 6]]7 [143561634], [14631236125]
[[12, 4, 3]]13 [20] [[12, 4, 5]]13 [18554], [184835595]
[[24, 18, 2]]17 [20] [[24, 18, 3]]17 [1332], [9511521119410532108411631]
[[21, 7, 2]]19 [16] [[21, 7, 3]]19 [19999998], [11313131313131319999991]
The final table shows the set of codes we have found whose parameters do not
appear in the literature.
Table 8 : New QECCs
[[n, k, d]]q Polynomials
[[61, 41, 5]]3
[10011111001], [1100101022020121010222110210221110202120101102
020022]
[[52, 7, 6]]3 [12211120111], [100000202100102111]
[[52, 35, 4]]3 [110000012], [11020221221200111012111020100112002220020221]
[[52, 15, 5]]3 [12010202], [12200202101110221121201]
[[40, 11, 6]]3 [112100201012],[11112000221100111020001]
[[40, 22, 5]]3 [1222100011], [12220111102211221220021121211111]
[[30, 2, 8]]3 [101101000202101],[10211201020112201]
[[32, 2, 7]]3 [1211021211201002],[122010112001001202]
[[30, 10, 4]]3 [120212022101],[1211111102001211202201]
[[30, 16, 4]]3 [11122111], [110112012020020210211011]
[[33, 1, 10]]3 [12000021012021021],[111000222111222222]
[[35, 4, 6]]3 [12122221121022], [101020002212012112]
[[36, 1, 8]]3 [11120020102002111],[100110211221022002]
[[36, 2, 8]]3 [12212100100121221],[1200122110112210021]
[[36, 18, 4]]3 [1110220111], [1102010202122112212020102011]
[[20, 4, 5]]3 [112212211], [1122201022211]
[[23, 1, 8]]3 [100202011222],[1202121101001]
[[26, 9, 6]]3 [111122121], [112200100222020102]
[[28, 0, 9]]3 [122221000112122],[122221000112122]
[[28, 12, 6]]3 [111000121], [121100111020121002111]
[[14, 2, 5]]3 [1101011], [111202111]
[[11, 1, 5]]5 [124114], [1122031]](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-37-320.jpg)
![GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 19
[[12, 4, 4]]5 [14102], [140124402]
[[16, 1, 6]]5 [1320214], [11112222]
[[20, 4, 6]]5 [121201212], [1231323013112]
[[30, 20, 4]]5 [142241], [14401243413033031434210441]
[[10, 2, 4]]7 [12134], [1230661]
[[14, 6, 4]]7 [12056], [12462526421]
[[40, 28, 4]]7 [1455541], [11152236066630031463035562341504546]
[[12, 2, 5]]13 [154816], [1582412411]
5.2. New Quantum Codes from Constacyclic Random Search. In this
section, we introduce another method to obtain quantum codes from classical codes
that we call constacyclic random (CR) search. We have obtained several new quan-
tum codes (codes with higher minimum distances than the ones in the literature)
from this search.
There is a small difference between the CR search and the earlier method of
constacyclic quantum search. In the CR search, we first find all g(x)|(xn
− a), then
find a large number of random polynomials f(x), and construct CSS codes with
g(x) and g(x)f(x).
By this method, we obtained a new [[11, 1, 5]]5-code which was also found by the
earlier search method and listed in Table 7. Additionally, we found a [[19, 1, 9]]17-
code from the generators g = x9
+4x8
+15x7
+15x6
+6x5
+x4
+12x3
+2x2
+3x+16
and gf = x10
+3x9
+11x8
+8x6
+12x5
+11x4
+7x3
+x2
+13x+1, which improves the
minimum distance of the known QECC [[19, 1, 5]]17-code in [23] by 4 units. Finally,
we have found a [[30, 10, 3]]19-code from the generators g = x4
+11x3
+14x2
+3x+18
and gf = x14
+ 11x12
+ 6x11
+ 8x10
+ x9
+ 4x8
+ 17x7
+ 4x6
+ x5
+ 13x4
+ 7x3
+
17x2
+ 15x + 8, which improves the minimum distance of the known QECC code
with parameters [[30, 10, 2]]19 given in [16] by 1 unit.
Our computational results suggest that this is a promising search method that
might yield even more new codes.
Acknowledgments
This work was supported by Kenyon College Summer Science Scholars program.
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Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022
Email address: aydinn@kenyon.edu
Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022
Email address: guidotti1@kenyon.edu
Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022
Current address: Department of Mathematics, University of Michigan, Ann Arbor, Michigan,
48104
Email address: paulliu@umich.edu](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-40-320.jpg)
![Contemporary Mathematics
Volume 785, 2023
https://doi.org/10.1090/conm/785/15772
Evaluation of iterated Ore polynomials
and skew Reed-Muller codes
Nabil Bennenni and André Leroy
Dedicated to Professor Syed Tariq Rizvi for his contribution to Ring and Module Theory.
Abstract. In this paper, we study two ways of evaluating iterated Ore poly-
nomials. We provide many examples and compare these evaluations. We use
the evaluation maps to construct Reed-Muller codes and compute explicitly
some of the data that are associated to such codes.
Introduction
The evaluation of polynomials is a keystone in many branches of mathematics.
In noncommutative algebra one of the most attractive notion of polynomials is the
Ore polynomials (a.k.a. skew polynomials). There are plenty of works dealing with
the evaluations of these polynomials with coefficients in division rings or prime
rings [7],[8]. In coding theory the evaluation of polynomials is used, in particular,
for constructing Reed-Muller codes [10], [5], [4].
Let us recall the general definition of the skew polynomial ring. Let K be a ring,
σ ∈ End(K), and δ a skew σ-derivation on K (i.e. δ is an additive map from K to K
such that for any a, b ∈ K, δ(ab) = σ(a)δ(b)+δ(a)b). The elements of a skew polyno-
mial ring R = K[t; σ, δ] are polynomials
n
i=0 aiti
with coefficients ai ∈ K written
on the left. The addition of these polynomials is the usual one and the multiplica-
tion is based on the commutation rule ta = σ(a)t+δ(a), for any a ∈ K. With these
two operations, R becomes a ring. When K is a division ring, R is a left principal
ideal domain. If σ is an automorphism of the division ring K then R is also a right
principal ideal domain. This construction can be iterated: we consider a ring K, a
sequence of skew polynomial rings R1 = K[t1; σ1, δ1], where σ1 ∈ End(K) and δ1
is a σ1-derivation of K. We then build R2 = R1[t1; σ2, δ2] = K[t1; σ1, δ1][t2; σ2, δ2],
where σ2 ∈ End(R1) and δ2 is a σ2-derivation of R1. Continuing this construc-
tion we arrived at the iterated polynomial ring in n noncommutative variables
R = Rn = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn]. The evaluation of skew polyno-
mials has a very different behavior than the usual one for commutative variables.
Most of the literature is related to right evaluation of univariate (i.e. n = 1)
polynomials with coefficients in a division ring (e.g. [8], [7]). For the case when
the base ring K is not necessarily a division ring, the pseudo-linear transforma-
tions are very useful [7]. When K is a division ring the left evaluation was
2020 Mathematics Subject Classification. Primary 94B05, 11T71, 14G50; Secondary 16L30.
Key words and phrases. Evaluation of iterated Ore polynomials, skew Reed-Muller codes.
c
2023 American Mathematical Society
23](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-42-320.jpg)
![24 NABIL BENNENNI AND ANDRÉ LEROY
also considered even when σ is not onto [9]. While considering the evaluation
of a polynomial f(t1, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] at a point
(a1, . . . , an) ∈ Kn
, we are quickly facing a “noncommutative phenomenon”: the
ideal I = R(t1 − a1) + · · · + R(t2 − a2) can be equal to R. We will show how it is
possible to avoid this problem.
Reed-Muller codes over F2 were introduced by Reed in 1954 [10]. Kasami,
Lin and Peterson in [6] extended the definition to any finite field Fq. Reed-Muller
codes are very useful in cryptography. Indeed, they show good resistance to linear
cryptanalysis. The interested reader may consult [4], for more details. Reed-Muller
codes are amongst the simplest examples of the class of geometrical codes, which
also includes Euclidean geometry and projective geometry codes.
The paper is organized as follows. Section 2 presents two ways of evaluating
iterated polynomials in a general context. We provide many examples and compare
these evaluations. In Section 3, we apply the evaluation maps to construct Reed-
Muller codes and compute some of the data that are associated to such codes.
1. Polynomial maps in n variables
In this section, we define a notion of evaluation for polynomials in an iterated
Ore extension R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] over an arbitrary ring K
(with unity). Throughout the paper we require that σi(K) ⊂ K and δi(K) ⊂ K
for any 1 ≤ i ≤ n. We will need some facts about iterated polynomial rings.
Remark 1.1. (1) When δ = 0 we denote the Ore extension as R =
K[t; σ]. Similarly when σ = id, we write R = K[t; δ].
(2) We remark that if u ∈ K an invertible element and, for any x ∈ K, we
have σ(x) = Iu(x) = uxu−1
then K[t; σ; δ] = K[u−1
t; u−1
δ]. Notice that
if δ = 0, this gives a usual polynomial ring.
(3) If σ is an endomorphism of the ring K and x, a are elements in K, we
define δa(x) = ax − σ(x)a. This is a σ-derivation on K called the inner
derivation on K. Let us notice that in this case we have, for any x ∈ K
that (t − a)x = σ(x)(t − a) so that R = K[t; σ, δ] = K[t − a; σ].
(4) Combining the two previous remarks we have that, for any u, v ∈ K with
u an invertible element, K[t; Iu, δv] = K[u−1
(t − v)] is a usual polynomial
ring.
Let us briefly recall the definition of the evaluation of a polynomial f(t) ∈ R =
K[t; σ, δ] at an element a ∈ K. We define f(a) ∈ K to be the only element of K
such that f(t) − f(a) ∈ R(t − a). This means, in particular, that a ∈ K is a right
root of f(t) when t − a is right factor of f(t). We then introduce, for any i ≥ 0, a
map Ni defined by induction as follows:
N0(a) = 1, Ni+1(a) = σ(Ni(a))a + δ(Ni(a)).
This leads to a concrete formula for the evaluation of any polynomial f(t) =
n
i=0 biti
∈ R = K[t; σ, δ] at an element a ∈ K:
f(a) =
n
i=0
biNi(a).
Before defining the evaluation of iterated polynomials, we now provide a few
classical examples.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-43-320.jpg)
![EVALUATION OF ITERATED ORE POLYNOMIALS 25
Example 1.2.
(1) Consider the complex number C and the conjugation denoted by “−”. We
then construct the Ore extension: R = C[t; −]; the commutation rule is
here t(a + ib) = (a − ib)t, where a, b ∈ R. An element a ∈ C is a (right)
root of t2
+ 1 if N2(a) + 1 = 0 i.e. aa + 1 = 0. From this it is clear that
t2
+ 1 is an irreducible polynomial in R. On the other hand, it is easy
to check that t2
+ 1 is a central polynomial and we get that the quotient
R/(t2
+ 1) is isomorphic to the quaternion algebra H. On the other hand,
the roots of the polynomial t2
− 1 are exactly the complex numbers of
norm 1.
(2) Another important kind of Ore extension is obtained by presenting the
Weyl algebra A1 = k[X][Y ; id., d
dx ], where k is a field. The commutation
rule that comes up is thus Y X = XY + 1. In characteristic zero this
algebra is simple (it doesn’t have any two sided ideal except 0 and A1). If
char(k) = p 0, A1 is a p2
dimensional algebra over its center k[Xp
, Y p
].
Evaluating powers of Y at X we get for instance Y 2
(X) = X2
+ 1,
Y 3
(X) = X3
+ 3X, and Y 4
(X) = X4
+ 6X2
+ 3.
We can iterate the procedure and get the Weyl algebra An(k).
(3) In coding theory, Ore extensions of the form Fq[t; θ], where q = pn
and
θ is the Frobenius automorphism defined by θ(a) = ap
, for a ∈ Fq, have
been extensively used (cf. [2]). In [7], both the evaluation and the factor-
ization for these extensions were described in term of classical (untwisted)
evaluation of polynomials. General Ore extensions have also been used
for constructing codes [3].
Let f(t1, t2, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] and an n-tuple
(a1, a2, . . . , an) ∈ Kn
be given (K is a ring which is assumed to be stable by the
maps σi and δi, with1 ≤ i ≤ n). We want to define the value of f(a1, a2, · · · , an).
It will be convenient to have notations for intermediate Ore extensions. So we put
R0 = K, and, for i ∈ {1, . . . , n − 1}, Ri = Ri−1[ti; σi, δi]. Notice that Rn = R.
Let (a1, a2, · · · , an) ∈ Kn
and consider f0 = f(t1, . . . , tn) ∈ R = Rn−1[tn; σn, δn].
We can define f1 = f(t1, . . . , tn−1, an) ∈ Rn−1 as the remainder of the division of
f0 on the right by tn − an. This is
f0 = q(t1, . . . , tn−1)(tn − an) + f1.
By the same procedure we divide f1 = f(t1, . . . , tn−1, an) by tn−1 − an−1 and get
the remainder f2 = f(t1, . . . , tn−2, an−1, an) ∈ Rn−2. This is
f1 = f(t1, . . . , tn−1, an) = q2(t1, t2, . . . , tn−1)(tn−1 − an−1) + f2.
Continuing this process, we define f3 ∈ Rn−3, . . . fn−1 ∈ R1, fn ∈ R0 = K. The
evaluation of f(t) at (a1, . . . , an) is fn ∈ K.
This leads to the following definition.
Definition 1.3. For f(t1, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn]
and (a1, . . . , an) ∈ Kn
, we define f(a1, . . . , an), the evaluation of f(t1, t2, . . . , tn) at
(a1, . . . , an), as the representative in K of f(t1, t2, . . . , tn) modulo In(a1, . . . , an) =
R1(t1 −a1)+· · ·+Rn−1(tn−1 −an−1)+R(tn −an), where, for 1 ≤ i ≤ n, Ri stands
for Ri = K[t1; σ1, δ1] · · · [ti; σi, δi].
Remark that In = In(a1, . . . , an) is an additive subgroup of R, + and is not, in
general, an ideal in R = Rn.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-44-320.jpg)
![26 NABIL BENNENNI AND ANDRÉ LEROY
Another way to compute the evaluation is to remark that the polynomials in
R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] can be written in a unique way as sums
of monomials of the form αl1,...,ln
tl1
1 tl2
2 · · · tln
n , for some 0 ≤ l1, l2, . . . , ln ≤ n and
αl1,...,ln
∈ K. The sum
n
i=1 li is called the degree of the monomial and the degree
of a polynomial is given by the degree of the monomials with higher degree. First
remark that if a monomial m = m(t1, . . . , tn) is of degree l, then for any a ∈ K∗
,
ma is a polynomial of degree l as well. We define the evaluation by induction on
the degree. In practice, it is sufficient to define the evaluation of a monomial.
For (a1, a2, . . . , an) ∈ Kn
and m = m(t1, . . . , tn) we define m(a1, . . . , an) as
follows:
If deg(m) = 1 and m = αti for some 1 ≤ i ≤ n and α ∈ K then m(a1, . . . , an) =
ai. So, assume that the evaluation of the monomials of degree l has been defined
and consider a monomial m = m(t) such that deg(m) = l and m = m
(t1, . . . , tj)tj
for some 1 ≤ j ≤ n, then m(a1, . . . , an) = m
(a1, . . . , aj) where the polynomial
m
(t1, . . . , tj) = m
(t1, . . . , tj)aj is of degree smaller than l.
Let us make some remarks about this evaluation.
Remark 1.4.
(1) First let us remark that, before we evaluate a polynomial, we must write
it with the variables appearing in the precise order t1, t2, . . . , tn (from left
to right). In other words, before evaluating a polynomial we must write
it as a sum of monomials of the form tl1
1 tl2
2 · · · tln
n .
(2) Of course, this is not the only possible definition, but although it might
look a bit strange, still it is natural if we want the zeros being right roots.
Let us look more closely to the case of two variables. In other words it is
natural for (a1, a2) to annihilate a polynomial of the form g(t1)(t2 − a2);
but we don’t necessarily expect (a1, a2) to be a zero of a polynomial of
the form (t1 − a1)h(t1).
(3) Since the base ring is not assumed to be commutative we must be very
cautious while evaluating a polynomial, even when the variables com-
mute. With this definition, t1t2 ∈ K[t1, t2] evaluated at (a, b) gives
(t1t2)(a, b) = ba. This might look very strange but if we think of evalua-
tion in terms of “operators” via the right multiplication by b followed by
the right multiplication by a this evaluation looks perfectly fine and the
apparent awkwardness disappears.
(4) We assumme that the different endomorphisms σis are such that σi(K) ⊆
K. In other words, for i 1, σi is an extension of σi|K to Ri−1. Some com-
mutation relations exist between the different endomorphisms σi. For in-
stance, consider the polynomial ring extension R = K[t1; σ1][t2; σ2]. If we
put σ2(t1) =
l
i=0 aiti
computing σ2(σ1(a)t1) = σ2(t1a) = σ2(t1)σ2(a),
leads to the following equations
∀ 0 ≤ i ≤ l, aiσi
1(σ2(a)) = σ2(σ1(a))ai.
Let us now give some examples.
Example 1.5.
(1) Let A1(k) = k[X][Y ; id., d
dX ] and (a, b) ∈ K2
, then
• Y X = XY + 1 = X(Y − b) + bX + 1 = X(Y − b) + b(X − a) + ba + 1
and hence (Y X)(a, b) = ba + 1.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-45-320.jpg)
![EVALUATION OF ITERATED ORE POLYNOMIALS 27
• Y X2
= X2
Y + 2X = X2
(Y − b) + bX2
+ 2X = X2
(Y − b) + bX(X −
a) + baX + 2(X − a) + 2a and hence (Y X2
)(a, b) = ba2
+ 2a.
• Y 2
X = XY 2
+ 2Y = XY (Y − b) + bX(Y − b) + bXb + 2(Y − b) + 2b
and hence (Y 2
X)(a, b) = b2
a + 2b.
(2) Consider the double Ore extension R = Fq[t1; θ][t2; θ], where q = pn
,
θ(a) = ap
, and θ(t1) = t1. A polynomial p(t1, t2) ∈ R can be written as
p(t1, t2) =
n
i=0 pi(t1)ti
2 =
i,j ai,jtj
1ti
2 and we can easily check that
p(t1, t2)(a, b) =
i,j
θj
(Ni(b))Nj(a) = b
(pi−1)pj
p−1 a
pi−1
p−1 .
Let us now turn to another possible way of evaluating a polynomial f(t1, . . . , tn)
∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] at (a1, a2, . . . , an) ∈ Kn
. We consider
the element of K representing f in the quotient R/I where I = R(t − a1) + R(t −
a2) + · · · + R(t − an). The set I is the usual left ideal of R and this evaluation
looks more classical. We now compare the two evaluations by comparing In =
R1(t1 − a1) + · · · + Rn−1(tn−1 − an−1) + R(tn − an) and I.
Theorem 1.6. Let K be a ring and R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn].
We consider (a1, a2, . . . , an) ∈ Kn
and put I = R(t−a1)+R(t−a2)+· · ·+R(t−an)
and In = R1(t1−a1)+· · ·+Rn−1(tn−1−an−1)+R(tn−an), where, for 1 ≤ i ≤ n, Ri,
Ri = K[t1; σ1, δ1] · · · [ti; σi, δi]. With these notations, the following are equivalent:
(1) In = I;
(2) R(ti − ai) ⊆ In;
(3) I = R;
(4) For 1 ≤ i j ≤ n, we have tj(ti − ai) ∈ In;
(5) For 1 ≤ i j ≤ n, we have σj(ti − ai)aj + δj(ti − ai) ∈ In;
(6) For 1 ≤ i j ≤ n, we have (tjti)(a1, . . . , an) = σj(ai)aj + δj(ai).
Proof. 1) ⇔ 2) is clear.
2) ⇒ 3) If I = R, then In = R and, for 1 ≤ i ≤ n, there exist polynomials
gi(t1, t2 · · · , ti) ∈ Ri such that 1 =
n
i=1 gi(t1, . . . , ti)(ti − ai). The change of
variables defined by putting yi = ti − ai gives that, for some hi ∈ Ri, we have
1 =
n
i=1 hi(y1, . . . , yi)yi. A comparison of the coefficients of degree zero of this
equality leads to a contradiction.
3) ⇒ 4) There exists c ∈ K such that tj(ti − ai) − c ∈ In ⊆ I. Since tj(ti − ai) ∈ I
we obtain c ∈ I hence, by 3), we must have c = 0. This shows that tj(ti − ai) ∈ In.
4) ⇒ 5) The fact that tj(ti − ai) ∈ In, implies that σj(ti − ai)tj + δj(ti − ai) ∈ In.
Since (tj − aj) ∈ In, we obtain that σj(ti − ai)aj + δj(ti − ai) ∈ In.
5) ⇒ 6) This is clear since 5) implies that tjti − σj(ai)aj − δj(ai) = σj(ti)tj +
δj(ti) − σj(ai)aj − δj(ai) ∈ In.
6) ⇒ 1) The equality in 6) implies that tjti−σj(ai)aj −δj(ai) ∈ In for 1 ≤ i j ≤ n.
This gives that tj(ti −ai) ∈ In. Since also we have that Ri(ti −ai) ⊆ In we conclude
that tj(ti − ai) ∈ In for any integer 1 ≤ i ≤ j ≤ n and hence R(ti − ai) ⊆ In for
any 1 ≤ i ≤ n. This yields In = I, as required.
Remark 1.7.
(1) If n = 1, we obviously have I = I1 and all points are good.
(2) In general, the two additive subset In ⊂ I are different. As mentioned
above, we will use In for our evaluation. The reason is that while eval-
uating with respect to I we often face the following problem: the left](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-46-320.jpg)
![28 NABIL BENNENNI AND ANDRÉ LEROY
R-module I can be the entire ring. So that the evaluation of any poly-
nomial at (a1, a2 . . . , an) ∈ Kn
with respect to I is zero. This the case
in the Weyl algebra R = A1(K) = K[t1][t2; id, d
dt1
] for the piont (0, 0)
since we then have t2t1 − t1t2 = 1 and hence Rt1 + Rt2 = R. This is not
the case with our evaluation since, for instance, t2t1 = t1t2 + 1, so that
t2t1 + I2(0, 0) = 1 + I2(0, 0) and hence the evaluation of t2t1 at (0, 0) is
just 1.
(3) In fact, it is quite often the case that I = R, even if we are using Ore
polynomials with zero derivations. Consider for instance the Ore extension
R = K[t1; σ1][t2; σ2] where K is a field and σ2 is an endomorphism of
K[t1; σ1] such that σ2(t1) = t1. It is easy to check that for any (a1, a2) ∈
K2
we have (t2 − σ1(a2))(t1 − a1) + (−t1 + σ2(a1))(t2 − a2) = σ1(a2)a1 −
σ2(a1)a2. So that if σ1(a2)a1−σ2(a1)a2 = 0, then the left ideal I(a1, a2) =
R. This shows that very often the evaluation modulo I turns out to be
trivial. Once again, this is not the case with our evaluation, since we have
that t2(t1 − a1) is represented by σ1(a2)a1 − σ2(a1)a2 modulo I2(a1, a2).
Definition 1.8. A point (a1, . . . , an) ∈ Kn
will be called a good point if
the two ways of evaluating a polynomial inK[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] at
(a1, . . . , a2) coincide i.e. if In(a1, . . . , an) = I.
The advantage of the good points is that in this case the evaluations via a left
ideal I and In coincide. But of course, we can still evaluate a polynomial at any
points via our additive subset In = R1(t1−a1)+· · ·+Rn−1(tn−1−an−1)+R(tn−an).
Example 1.9. (1) In the classical case (σi = idK and δi = 0, for every
1 ≤ i ≤ n), every point (a1, . . . , an) ∈ Kn
is good.
(2) If K is a division ring σ1 = idK, δ1 = 0 and σ2 = id, δ2 = d/dt1, we have
for any a, b ∈ K, (t2 − b)(t1 − a) = (t1 − a)(t2 − b) + 1. This shows that
in this case there are no good points.
Working with In instead of I we avoid the problem of having points that are
zeroes of every polynomial in the ring R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn].
We will consider Reed-Muller codes and hence we will need to evaluate polyno-
mials in several variables. We consider the case of iterated Ore polynomials defined
on a finite base ring (field) K. In classical Reed-Muller coding the polynomials that
are used to make evaluations are restricted to be of some bounded degree in each
variables. Indeed polynomial maps associated to Xq
− X ∈ Fq[X] are identically
zero. So our first task is to consider what is the analogue of this polynomial in an
Ore polynomial ring. We will need a few definitions.
Definition 1.10. Let a ∈ K be an element of a division ring K, σ and δ an
endomorphism of K and a σ-derivation of K, respectively.
(1) For a nonzero x ∈ K, we denote ax
= σ(x)ax−1
+ δ(x)x−1
and put
Δ(a) = {ax
| x ∈ K∗
}. This set is called the (σ, δ) conjugacy class of a.
(2) The map Ta : K −→ K defined by Ta(x) = σ(x)a + δ(x) is called the
(σ, δ) pseudo-linear map associated to a.
(3) The (σ, δ) centralizer of a is the set Cσ,δ
(a) = {x ∈ K∗
| ax
= a} ∪ {0}.
Let us mention that the notion of σ, δ conjugation appears naturally due to
the fact that while evaluating a product fg we have the nice formula fg(a) = 0
if g(a) = 0 and (fg)(a) = f(ag(a)
)g(a) if g(a) = 0. Let us also mention that the](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-47-320.jpg)
![EVALUATION OF ITERATED ORE POLYNOMIALS 29
evaluation at an element a ∈ K is strongly related to the (σ, δ)-pseudo linear map Ta
via the equality f(a) = f(Ta)(1) or more generally that, for any h ∈ R = K[t; σ, δ],
a ∈ K and x ∈ K∗
, h(Ta)(x) = h(ax
)x. We first need the following lemma which
is part of folklore.
Theorem 1.11. Let a be an element of a division ring K and f(t) ∈ R =
K[t; σ, δ] be a polynomial of degree n. Then:
(1) Cσ,δ
(a) is a subdivision ring of K.
(2) The map Ta is a right Cσ,δ
(a) linear map.
(3) f(t) has roots in at most n (σ, δ)-conjugacy classes, say {Δ(a1), . . . , Δ(ar)},
r ≤ n;
(4)
r
i=1 dimC(ai) ker(f(Tai
)) ≤ n, where C(ai) := Cσ,δ
(ai) for 1 ≤ i ≤ r.
Proof. We refer the reader to [8] and [11].
Notice that point (4) above generalizes several classical theorems, in particular,
the theorem by Gordon-Motzkin that states that a polynomial f(X) ∈ K[X] with
coefficients in a division ring K can have roots in at most n = deg(f) classical
(σ = id., δ = 0)-conjugacy classes. Theorem 1.12 below can be found in [7].
Theorem 1.12. Let p be a prime number and Fq be the finite field with q = pn
elements. Denote by θ the Frobenius automorphism. Then:
(a) There are p distinct θ-conjugacy classes in Fq.
(b) Cθ
(0) = Fq and, for 0 = a ∈ Fq, we have Cθ
(a) = Fp.
(c) In Fq[t; θ], the least left common multiple of all the elements of the form
t − a for a ∈ Fq is the polynomial G(t) := t(p−1)n+1
− t. In other words,
G(t) ∈ Fq[t; θ] is of minimal degree such that G(a) = 0 for all a ∈ Fq.
(d) The polynomial G(t) obtained in c) above is invariant, i.e. RG(t) =
G(t)R.
Proof.
(a) Let us denote by g a generator of the cyclic group F∗
q := Fq {0}. The
θ-conjugacy class determined by the zero element is reduced to {0} i.e. Δ(0) = {0}.
The θ-conjugacy class determined by 1 is a subgroup of F∗
q: Δ(1) = {θ(x)x−1
| 0 =
x ∈ Fq} = {xp−1
| 0 = x ∈ Fq}. It is easy to check that Δ(1) is cyclic generated
by gp−1
and has order pn
−1
p−1 . Its index is (F∗
q : Δ(1)) = p − 1. Since two nonzero
elements a, b are θ-conjugate if and only if ab−1
∈ Δ(1), we indeed get that the
number of different nonzero θ-conjugacy classes is p − 1. This yields the result.
(b) If a ∈ Fq is nonzero, then Cθ
(a) = {x ∈ Fq | θ(x)a = ax} i.e. Cθ
(a) = Fp.
(c) We have, for any x ∈ Fq, (t(p−1)n+1
− t)(x) = θ(p−1)n
(x) . . . θ(x)x − x.
Since θn
= id, and Nn(x) := θn−1
(x) . . . θ(x)x ∈ Fp, we get (t(p−1)n+1
− t)(x) =
x(θn−1
(x) . . . θ(x)x)p−1
− x = xNn(x)p−1
− x = 0. This shows that indeed G(t)
annihilates all the elements of Fq and hence G(t) is a left common multiple of the
linear polynomials {(t − a) | a ∈ Fq}. Let h(t) := [t − a | a ∈ Fq]l denote their
least left common multiple. It remains to show that deg h(t) ≥ n(p − 1) + 1. Let
0 = a0, a1, . . . , ap−1 ∈ Fq be elements representing the θ-conjugacy classes (Cf. a)
above). Denote by C0, C1, . . . , Cp−1 their respective θ-centralizer. The formulas
recalled in the paragraph before Theorem 1.11 shows that h(Ta)(x) = h(ax
)x =
0 for any nonzero element x ∈ Fq and any element a ∈ {a0, . . . , ap−1}. Hence
ker h(Tai
) = Fq for 0 ≤ i ≤ p − 1. Using Inequality (4) in Theorem 1.11 and](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-48-320.jpg)
![30 NABIL BENNENNI AND ANDRÉ LEROY
the statement b) above, we get deg h(t) ≥
p−1
i=0 dimCi
ker h(Tai
) = dimFq
Fq +
p−1
i=1 dimFp
Fq = 1 + (p − 1)n, as required.
(d) Since θn
= id., we have immediately that G(t)x = θ(x)G(t) and obviously
G(t)t = tG(t).
Let us now consider an iterated Ore extension K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn],
where K is a finite ring. Using classical notations from the world of algebraic ge-
ometry, we consider
I(Kn
) = {f(t1, . . . , tn) ∈ R | ∀(a1, . . . , an) ∈ Kn
, f(a1, . . . , an) = 0}.
In other words: I(Kn
) = ∩(a1,...,an)In(a1, . . . , an). In the case of an Ore extension
Fq[t1; θ] . . . [tn; θ], where θ is the Frobenius automorphism of Fq = Fpn , the above
theorem shows that, for any 1 ≤ i ≤ n, t
(n−1)p
i − ti ∈ I(Kn
). Having this in
mind we introduce, for any 1 ≤ i ≤ n, the monic polynomials Gi = Gi(t1, . . . ti)
of minimal degree in Ri = K[t1; σ1, δ1] . . . [ti; σi, δi] such that Gi ∈ I(Kn
). With
these notations, we can now state the following proposition.
Proposition 1.13. Let K be a finite ring and, for 1 ≤ i ≤ n, consider the
polynomials Gi ∈ Ri = K[t1, σ1, δ1] . . . [ti, σi, δi] defined above. Then
I(Kn
) = R1G1(t1) + R2G2(t1, t2) + · · · + RnGn(t1, . . . , tn).
Proof. It is clear that R1G1(t1) + R2G2(t1, t2) + · · · + RnGn(tn) ⊆ I(Kn
).
Now, if a polynomial f ∈ K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] belongs to I(Kn
),
then, for any (a1, . . . , an) ∈ Kn
, f(a1, a2, . . . , an) = 0. Let us write the consecutive
remainders obtained during the evaluation process as follows. For any (a1, . . . , an) ∈
Kn
, we have
f(t1, . . . , tn) = q1(t1, . . . , tn)(tn − an) + f1(t1, . . . , tn−1, an).
f1(t1, . . . , tn−1, an) = q2(t1, . . . , tn−1)(tn−1 − an−1) + f2(t1, . . . , tn−2, an−1, an).
Continuing this process, we get
fn−2(t1, t2, a3, . . . , an) = qn−1(t1, t2)(t2 − a2) + fn−1(t1, a2 . . . , an) and
fn−1(t1, a2 . . . , an) = qn(t1)(t1 − a1) + fn(a1, . . . , an−1, an).
Since f(a1, a2, . . . , an) = fn(a1, . . . , an−1, an) = 0, we conclude that for any for
any (a1, . . . , an ∈ Kn
, fn−1(t1, a2 . . . , an) ∈ R1(t1−a1), i.e. that fn−1(t1, a2 . . . , an)
∈ R1G1(t1). Similarly, we have that fn−2(t1, t2, a3, . . . , an) − fn−1(t1, a2 . . . , an) ∈
R2 ∩ I(Kn
) = R2G2(t1, t2). An easy induction yields that for any 1 ≤ i ≤ n,
we have fn−i(t1, . . . ti, ai+1, . . . , an) − fn−i+1(t1, . . . , ti−1, ai, . . . , an) ∈ RiGi. From
this we get successively that f ∈ RnGn + f1 ⊆ RnGn + Rn−1Gn−1 + f2 ⊆ RnGn +
Rn−1Gn−1 + Rn−3Gn−3 + f3 ⊆ R1G1(t1) + R2G2(t1, t2) + · · · + RnGn(t1, . . . , tn).
This yields the desired result.
2. Skew Reed-Muller codes using iterated skew polynomial rings
In the first part of this section, we give some preliminaries on Reed-Muller
codes over the commutative polynomial ring.
Let m be a positive integer and let P1, P2, . . . , Pn be the n = qm
points in
the affine space Am
(Fq) . For any integer r with 0 ≤ r ≤ m(q − 1), let Rm ⊆
Fq [x1, x2, . . . , xm] /(xp
1 − x1, xp
2 − x2, · · · , xp
m − xm) be the set representing polyno-
mials of degree less or equal than r.](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-49-320.jpg)
![EVALUATION OF ITERATED ORE POLYNOMIALS 31
Definition 2.1. Let f =
n
i=0 aIXI
be a polynomial in Fq [x1, x2, . . . , xm],
where for I = (i1, . . . , in) we define XI
to be xi1
xi2
· · · xin
. The operator evaluation
ev(f) at b = (b1, b2 . . . bn) ∈ Fn
q is defined as
Rm → Fq
ev : f =
i∈I
aiXi
→ f(b) =
i∈I
ai(bi
).
The Reed-Muller codes of the length n = qm
can be obtained as evaluation of
the polynomial f in Rm in all points P1, P2, · · · , Pn in the affine sapce Am
(Fq) .
The Reed-Muller codes of order r is defined as
Rq(r, m) = {(f (P1) , f (P2) , . . . , f (Pn)) | f ∈ Rm} = {ev(f) : f ∈ Rm, deg f ⩽ r} .
The function ev : Rm → Fq is an homomorphism of algebras (see [1]). Hence
the Reed-Muller codes is generated by the codewords ev(f) where f is a monomial
of degree less than or equal to r in the set of monomials in m variables. It will be
denoted
M = {Xa1
1 Xa2
2 . . . Xan
n } .
2.1. Iterated skew polynomial and Reed-Muller codes. In a recent pa-
per, skew Reed-Muller codes were defined via the use of iterated Ore extensions
with three variables over a finite field [5]. The authors used inner derivations and
inner automorphisms, and (cf. Remarks 1.1) these can be erased by a change of
variables. Their idea was to used Gröbner bases to make computations. They used
the available software on the subject. But their methods used the ideal I in our
notations and hence they faced the problem that I might be equal to the whole
iterated Ore extension R. Some of their computations were wrong and for the sake
of future works, we will briefly correct them. In the lines below the variables Xi
are the one used in their paper.
1. Let F4 = F2(α), where α2
= α + 1. In the ring R1 = F4 [Y1; θ] where θ is
the Frobenius automorphism, we put X1 = Y1 + 1 and we have the commutation
relation
X1α = (Y1 + 1)α = θ(α)Y1 + α = θ(α)X1 + θ(α) + α = α2
X1 + 1.
2. In the ring R2 = R1 [Y2] we put X2 = α2
Y2 + Y1 + α2
so that we have R2 =
R1[X2; θα, δX1+α], where θα(x) = α−1
xα for any x ∈ R1. In R2 we have the above
commutation relation X1α = α2
X1 + 1 together with
X2α = αX2 + X1 + 1 and X2X1 = αX1X2 + α2
X2 + α2
X2
1 + α2
.
3. In the ring R3 = R2[Y3], we put X3 = α2
Y3 + αY1 + α. We then have R3 =
R2 [X3; θα, δαX1
] where, as above θα(x) = α−1
xα, for any x ∈ F4. We have the
above commutation relations together with
X3α = αX3 + αX1 + α,
X3X1 = αX1X3 + α2
X3 + α2
X2
1 + α2
X1,
X3X2 = X2X3 + α2
X1X3 + α2
X3 + X1X2 + αX2
1 + α2
X1 + X2 + 1.
Of course, the monomials in X1, X2, X3 are not monomials in Y1, Y2, Y3 and con-
versely. The advantage of the variables Y1, Y2, Y3 is clear. We continue with these
variables and consider,](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-50-320.jpg)
![32 NABIL BENNENNI AND ANDRÉ LEROY
Rn = Rn−1[Yn] = R1[Y2, Y3, . . . , Yn] = F4[Y1, θ][Y2, Y3, . . . , Yn].
As in the commutative case, there are polynomials that annihilate any element of
Fn
q . Thanks to Theorem 1.12 and 1.13, we have that, in our situation I(Kn
) =
R1G1 + R2G2 + · · · + RnGn where G1 = Y 3
1 − Y1 and Gi = Y 4
i − Yi for any
2 ≤ i ≤ n (notice that the form of G1 is due to the theorem 1.12). This iterated
skew polynomial ring leads to a new family of skew Reed-Muller code generated by
the evaluations of a base of monomials in m variables and of degree less than or
equal to r given by
M =
Y l1
1 Y l2
2 . . . Y ln
n
.
By iterated skew polynomial rings given above we have the evaluation of f ∈ Rm
at b = (b1, b2 . . . bn) ∈ Fn
q is defined as
Rm → Fq
ev : f(X1, X2, . . . , Xn) → f(b1, . . . , bn) =
n
i=0
P(b2, . . . , bn)θ
i(i+1)
2 (b1).
Where f(X1, X2, . . . , Xn) = g(Y1)P(Y2, . . . , Yn) ∈ R1G1 + R2G2 + · · · + RnGn. By
the iterated skew polynomial ring and the evaluation we have the new new family
of the skew Reed-Muller codes in the following proposition.
Proposition 2.2. The skew Reed-Muller codes of parameters r and m over
Rm = Fq[X1; θ1, δ1] . . . [Xm; θm, δm] are generated as
R(r, m)
def
=
n
i=0
P(b2, . . . , bn)θ
i(i+1)
2 (b1) : P ∈ Rm such that deg P ⩽ deg Gn
.
The choice of the rows used to form the matrix generating the Reed-Muller
codes R(r, m) consists in selecting the monomials Y l1
1 Y l2
2 . . . Y ln
n of degree less than
or equal to r = (n−1)p+1. This choice is based on the order of the monomials. The
group of permutations of Reed-Muller codes R(r, m) is the set of transformations
affines x → Ax + b where A ∈ Fm×m
q is an invertible matrix and b ∈ Fm
q .
2.2. Skew Reed-Muller codes using polynomial maps in n variables.
In this section we give the construction of skew Reed-Muller codes using polynomial
maps in n variables. We know that the Reed-Muller codes are defined as the codes
generated by the evaluations of a base of monomials in m variables and of degree
less than or equal to r, the set of monomials in m variables will be denoted
M =
tl1
1 tl2
2 . . . tln
n
.
Notice that in the case of iterated Ore extensions K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn]
it is always possible to order the indeterminates of a word in t1, . . . , tn so that the
only words we have to consider are the one of the form tl1
1 tl2
2 . . . tln
n . We now give
the evaluation of some monomials in R = K[t1, σ1, δ1][x2, σ2, δ2], · · · , [xn, σn, δn] at
(a1, a2, · · · , an) ∈ Kn
.
(1) Consider the polynomial ring R = K[t1; σ1, δ1][t2; σ2, δ2] and let us evalu-
ate the polynomials t1t2 at (a1, a2) ∈ K2
. We have t1t2 = t1(t2 − a2) +
t1a2 = t1(t2 − a2) + σ1(a2)t1 + δ1(a2). This leads to (t1t2)(a1, a2) =
σ1(a2)a1 + δ1(a2).](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-51-320.jpg)
![EVALUATION OF ITERATED ORE POLYNOMIALS 33
(2) Consider the polynomial ring R = K[t1; σ1, δ1][t2; σ2, δ2][t3; σ3, δ3] and let
us evaluate the polynomials t1t2t3 at (a1, a2, a3) ∈ K3
. We have t1t2t3 =
t1t2(t3 − a3) + t1t2a3 = t1t2(t3 − a3) + t1(σ2(a3)t2 + δ2(a3)) = t1t2(t3 −
a3)+t1σ2(a3)t2 +t1δ2(a3) = t1t2(t3 −a3)+σ1(σ2(a3))t1t2 +σ1(δ2(a3))t1 +
δ1(δ2(a3)). This leads to
(t1t2t3)(a1, a2, a3) = σ1(σ2(a3))a1a2 + σ1(δ2(a3))a1 + δ1(δ2(a3)).
(3) Consider the polynomial ring
R = K[t1; σ1, δ1][t2; σ2, δ2][t3; σ3, δ3][t4; σ4, δ4].
We have
(t1t2t3t4)(a1, a2, a3, a4) = σ1(σ2(σ3(a4)a3)a2)a1 + δ1(σ2(σ3(a4)a2)a3) +
σ1(δ2(σ3(a4)a3))a1 +δ1(δ2(σ3(a4)a3))+σ1(σ2(δ3(a4))a2)a1 +δ1(σ2(δ3(a4))a2)+
σ1(δ2(δ3(a4)))a1 + δ1(δ2(δ3(a4)))).
Polynomial maps in n variables and the evaluation leads to the new family of skew
Reed-Muller codes in the following example.
Example 2.3. In this example, we give the construction of the Reed-Muller
codes with polynomial maps in 2 or 3 variables. In these examples we used the
Frobenius automorphsim for σ1 and the identity for σ2 and σ3. The derivations are
all zeroes.
(1) The evaluation of the set monomial {1, t1, t2, t1t2} over
R = F4[t1; σ1, δ1][t2; σ2, δ2] gives the Reed-Muller code with parameters
[16, 4, 8].
(2) The evaluation of the set monomial {1, t1, t2, t2t1} over
R = F4[t1; σ1, δ1][t2; σ2, δ2] gives the Reed-Muller code with parmeters
[16, 4, 7].
(3) The evaluation of the set monomial {1, t1, t2, t1, t3, t1t3, t2t3, t1t2t3} over
R = F4[t1; σ1, δ1][t2; σ2, δ2][t3; σ3, δ3] gives the Reed-Muller code with pa-
rameters [64, 8].
References
[1] M. Bardet, V. Dragoi, A. Otmani, and J.P. Tillich, Algebraic properties of polar codes from
a new polynomial formalism, IEEE International Symposium on Information Theory (ISIT),
230–234, 2016.
[2] Delphine Boucher and Felix Ulmer, Coding with skew polynomial rings, J. Symbolic Comput.
44 (2009), no. 12, 1644–1656, DOI 10.1016/j.jsc.2007.11.008. MR2553570
[3] M’Hammed Boulagouaz and André Leroy, (σ, δ)-codes, Adv. Math. Commun. 7 (2013), no. 4,
463–474, DOI 10.3934/amc.2013.7.463. MR3119685
[4] I. Dumer, R. Fourquet, and C. Tavernier, Cryptanalysis of Block Ciphers via Decoding of Long
Reed-Muller Codes, Algebraic and Combinatorial Coding Theory (ACCT 2010), Novosibirsk,
Russia September 2010.
[5] Willi Geiselmann and Felix Ulmer, Skew Reed-Muller codes, Rings, modules and codes, Con-
temp. Math., vol. 727, Amer. Math. Soc., [Providence], RI, [2019] c
2019, pp. 107–116, DOI
10.1090/conm/727/14628. MR3938143
[6] Tadao Kasami, Shu Lin, and W. Wesley Peterson, New generalizations of the Reed-
Muller codes. I. Primitive codes, IEEE Trans. Inform. Theory IT-14 (1968), 189–199, DOI
10.1109/tit.1968.1054127. MR275989
[7] André Leroy, Noncommutative polynomial maps, J. Algebra Appl. 11 (2012), no. 4, 1250076,
16, DOI 10.1142/S0219498812500764. MR2959425](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-52-320.jpg)
![34 NABIL BENNENNI AND ANDRÉ LEROY
[8] T. Y. Lam, A. Leroy, and A. Ozturk, Wedderburn polynomials over division rings. II, Non-
commutative rings, group rings, diagram algebras and their applications, Contemp. Math.,
vol. 456, Amer. Math. Soc., Providence, RI, 2008, pp. 73–98, DOI 10.1090/conm/456/08885.
MR2416145
[9] T. Y. Lam and André Leroy, Principal one-sided ideals in Ore polynomial rings, Algebra and
its applications (Athens, OH, 1999), Contemp. Math., vol. 259, Amer. Math. Soc., Providence,
RI, 2000, pp. 333–352, DOI 10.1090/conm/259/04106. MR1780532
[10] D.E. Muller, Application of Boolean algebra to switching circuit design and to error detection,
Transactions of the IRE professional group on electronic computers, vol.3 (1954), 6–12.
[11] André Leroy and Adem Ozturk, Algebraic and F-independent sets in 2-firs, Comm. Algebra
32 (2004), no. 5, 1763–1792, DOI 10.1081/AGB-120029901. MR2099700
USTHB, Faculté de Mathématiques, BP 32 El Alia Bab Ezzouar, Algiers, Algeria
Email address: nabil.bennenni@gmail.com
Université Artois, UR 2462, Laboratoire de Mathématiques de Lens (LML), F-62300
Lens, France
Email address: andre.leroy@univ-artois.fr](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-53-320.jpg)
![Contemporary Mathematics
Volume 785, 2023
https://doi.org/10.1090/conm/785/15773
Identities of the algebra O ⊗ O
Guy Blachar, Louis H. Rowen, and Uzi Vishne
To Professor Syed Tariq Rizvi, for his contribution to Ring and Module Theory
Abstract. We find nonassociative polynomial identities of minimal degree,
which is 7, for the algebras O⊗O and M2(O), where O is the octonion algebra.
1. Introduction
One of the basic invariants of an algebra A satisfying a polynomial identity (PI)
is the minimal degree of an identity, which we call the PI degree. A commutative
algebra has PI degree 2, because it satisfies [x1, x2]. That the PI degree of the
matrix algebra Mn(F) is 2n is the Amitsur-Levitzki theorem. The Grassmann
algebra satisfies [x1, [x2, x3]], so its PI degree is 3.
The same notion applies to nonassociative algebras. The substitution in an
associative polynomial is not well defined if the algebra is not associative, so an
identity of A would be an element in the free nonassociative algebra, which vanishes
under every substitution from A. Again we are concerned with the minimal degree
of an identity. The associator (x1, x2, x3) = (x1x2)x3 − x1(x2x3) is an identity in
the nonassociative variety precisely when the algebra is associative. The notion
of polynomial identities, and thus the PI degree, depends on the variety. If V is a
variety of algebras and A ∈ V, the nontrivial polynomial identities of A with respect
to V are nonzero elements of the free algebra in V, so one excludes consequences of
the defining identities of V.
Let O denote the split octonion algebra over a base field F. The alternative
PI degree of the octonion algebra is 5 (see Section 3 below), because the identities
of lower degree, namely the consequences of (x, x, y) and (y, x, x), hold in any
alternative algebra. The purpose of this note is to attract attention to the following
question:
Question 1.1. What is the (nonassociative) PI degree of the matrix algebras
Mn(O)? What is the PI degree of the tensor power O⊗n
?
Here we are concerned with the (general) nonassociative identities, as the cen-
tral simple algebras Mn(O) and O⊗n
are not alternative (for n 1); indeed, the
only simple alternative algebras are associative or the octonion algebras.
2020 Mathematics Subject Classification. Primary 17A30; Secondary 17D05, 16R99.
Partially supported by ISF grant #1994/20.
c
2023 American Mathematical Society
35](https://image.slidesharecdn.com/26756602-250523162257-cf7e36c6/85/Algebra-And-Coding-Theory-A-Leroy-S-K-Jain-54-320.jpg)
Algebra And Coding Theory A Leroy S K Jain
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- 5. 785 Algebra and Coding Theory Virtual Conference in Honor of Tariq Rizvi Noncommutative Rings and their Applications VII July 5–7, 2021 Université d’Artois, Lens, France Virtual Conference Quadratic Forms, Rings and Codes July 8, 2021 Université d’Artois, Lens, France A. Leroy S. K. Jain Editors
- 6. Algebra and Coding Theory Virtual Conference in Honor of Tariq Rizvi Noncommutative Rings and their Applications VII July 5–7, 2021 Université d’Artois, Lens, France Virtual Conference Quadratic Forms, Rings and Codes July 8, 2021 Université d’Artois, Lens, France A. Leroy S. K. Jain Editors
- 8. 785 Algebra and Coding Theory Virtual Conference in Honor of Tariq Rizvi Noncommutative Rings and their Applications VII July 5–7, 2021 Université d’Artois, Lens, France Virtual Conference Quadratic Forms, Rings and Codes July 8, 2021 Université d’Artois, Lens, France A. Leroy S. K. Jain Editors
- 9. EDITORIAL COMMITTEE Michael Loss, Managing Editor John Etnyre Angela Gibney Catherine Yan 2020 Mathematics Subject Classification. Primary 16-XX, 16N40, 16S34, 16S36, 11T71, 94BXX, 94B05, 94B15, 94B60. Library of Congress Cataloging-in-Publication Data Names: Rizvi, S. Tariq, honoree. | Leroy, André (André Gerard), 1955– editor. | Jain, S. K. (Surender Kumar), 1938– editor. | International Conference on Noncommutative Rings and Their Applications (7th : 2021 : Lens, France), author. Virtual Conference on Quadratic Forms, Rings and Codes (2021 : Lens, France) Title: Algebra and coding theory : virtual conference in honour of Tariq Rizvi, Noncommutative Rings and Their Applications, VII, July 5–7, 2021, Université d’Artois, Lens, France : Virtual Conference on Quadratic Forms, Rings and Codes, July 8, 2021, Université d’Artois, Lens, France / A. Leroy, S.K. Jain, editors. Description: Providence, Rhode Island : American Mathematical Society, [2023] | Series: Con- temporary mathematics, 0271-4132 ; volume 785 | Includes bibliographical references. Identifiers: LCCN 2022048347 | ISBN 9781470468590 (paperback) | ISBN 9781470472986 (ebook) Subjects: LCSH: Coding theory–Congresses. | Algebra–Congresses. | AMS: Associative rings and algebras. | Associative rings and algebras – Radicals and radical properties of rings – Nil and nilpotent radicals, sets, ideals, rings. | Associative rings and algebras – Rings and algebras arising under various constructions – Group rings. | Associative rings and algebras – Rings and algebras arising under various constructions – Ordinary and skew polynomial rings and semigroup rings. | Number theory – Finite fields and commutative rings (number-theoretic aspects) – Algebraic coding theory; cryptography. | Information and communication, circuits – Theory of error-correcting codes and error-detecting codes. | Information and communication, circuits – Theory of error-correcting codes and error-detecting codes – Linear codes, general. | Information and communication, circuits – Theory of error-correcting codes and error-detecting codes – Cyclic codes. | Information and communication, circuits – Theory of error-correcting codes and error-detecting codes – Other types of codes. Classification: LCC QA268 .A388 2023 | DDC 512/.4601154–dc23/eng20230415 LC record available at https://lccn.loc.gov/2022048347 Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: https://doi.org/10.1090/conm/785 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission@ams.org. c 2023 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18
- 10. This volume is dedicated to Syed Tariq Rizvi for his outstanding contri- butions to Mathematics and in particular to Rings and Modules. Tariq Rizvi, Ohio State University, Lima, Ohio.
- 12. Contents Preface ix List of participants xi A note on checkable codes over Frobenius and quasi-Frobenius rings R. R. de Araujo, A. L. M. Pereira, and C. Polcino Milies 1 Good classical and quantum codes from multi-twisted codes Nuh Aydin, Thomas Guidotti, and Peihan Liu 7 Evaluation of iterated Ore polynomials and skew Reed-Muller codes Nabil Bennenni and André Leroy 23 Identities of the algebra O ⊗ O Guy Blachar, Louis H. Rowen, and Uzi Vishne 35 When Is Every Non Central-Unit A Sum of Two Nilpotents? Simion Breaz and Yiqiang Zhou 47 Asymptotic Brauer p-dimension Adam Chapman and Kelly McKinnie 57 Five theorems on Gorenstein global dimensions Lars Winther Christensen, Sergio Estrada, and Peder Thompson 67 Quasi-Self-Dual Codes over a Non-Unital Ring of Order 4 Steven T. Dougherty and Serap Şahinkaya 79 Codes from the Skew Ring M2(F2) ϕ G S. T. Dougherty, Serap Şahinkaya, and Deniz Ustun 89 The underlying vector spaces of certain endomorphism rings Gabriella D’Este 107 Algebraic structures from the point of view of complete multiplicative lattices Alberto Facchini 113 Z2Z4-additive codes as codes over rings Cristina Fernández-Córdoba and Steve Szabo 133 On Utumi rings and continuous regular Baer rings Gangyong Lee, Cosmin S. Roman, Nguyen Khanh Tung, and Xiaoxiang Zhang 151 vii
- 13. viii CONTENTS Matrices representable as product of conjugates of a singular matrix S. K. Jain and André Leroy 163 Almost Projective Modules Over Non-Hereditary Algebras Fatma Kaynarca, Gabriella D’Este, and Derya Keskin Tütüncü 167 On prime ideals with generalized derivations in rings with involution Mohammad Salahuddin Khan, Adnan Abbasi, Shakir Ali, and Mohammed Ayedh 179 On nil algebras and a problem of Passman concerning nilpotent free algebras Jan Krempa 197 Rings with transitive chaining of idempotents Xavier Mary 205 On Continuous Hulls of Rings and Modules Jae Keol Park and S. Tariq Rizvi 213
- 14. Preface As an outgrowth of the two conferences ”Noncommutative rings and their Ap- plication (NCRA)” and ”Quadratic forms, rings and codes(QFRC)” held online in July 2021 in Lens (France), this volume contains a variety of topics in algebra and exhibits their interplay as presented in the meetings by the participants. The NCRA conference was dedicated to our colleague Tariq Rizvi. Tariq Rizvi is a prolific algebraist and his impact on rings and modules is tremendous. His influence is really impressive and many algebraists around the world benefited from his experience, expertise and generosity. He is frank, humble, generous and it is always a pleasure to meet him with his welcoming smile. We are lucky and proud to count him as our friend and we are very happy to give him, through this volume, a small token of our respect and gratitude. Some of the papers included in the volume reflect the influence of his work on the questions related to the direct sums of continuous and quasi-continuous modules, among others. One of the aims of these meetings was to establish connections between ring theory and coding theory. In particular, the applications to coding theory of Frobe- nius rings, the skew group rings and iterated Ore extensions are the subject of some of the papers. The questions on the classical topics, such as Utumi rings, Baer rings, nil and nilpotent algebras, Brauer groups are discussed in some papers. Other arti- cles deal with questions that relate to the elementwise study for rings and modules. Lastly, this volume includes papers dealing with questions in homological algebra and lattice theory. All these works show the vivacity of the research in the area of noncommutative rings and its influence on other subjects. As usual, the papers went through a strict refereeing process and we would like to use this opportunity to thank all the anonymous referees who kindly reviewed the papers. We thank Christine Thivierge who was so helpful and efficient in handling this publication. André Leroy and S.K. Jain ix
- 16. List of Participants Mehdi Aaghabali University of Edinburg Evrim Akalan Hacettepe University, Ankara. Mustafa Alkan Akademiz University, Antalya. Meltem Altun- Özarslan Hacettepe University, Ankara. Dilshad Alghazzawi Université d’Artois. Marı́a José Arroyo Universidad Autónoma Metropolitana, Iztapalapa (Mexico) Noha Abdelghany Colby College. Mona Abdi Shahrood University of Technology. Sami Alabiad King Saud University, Riyadh. Yusuf Alagoz Siirt University. Dilshad Alghazzawi Jeddah University, KAU. Shakir Ali Aligarh Muslim University, Aligarh. Abeer Ali Rajaallah Alharbi King Abdulaziz University Branch Rabigh Saudi Arabia. Anani Alkathiry Umm Al-Qura University. Yousef Alkhamees King Saud University. Meltem Altun Hacettepe University, Ankara. Mohammad Ashraf Alighar Muslim University. Pinar Aydogdu Hacettepe University, Ankara. Per Bäck Mälardalen University,. Ayman Badawi American University of Sharjah , Sharjah. Mhammed Boulagouaz Universite Sidi Mohammed ben Abdellah Fes. Ranya Djihad Boulanouar Algiers University USTHB, Alger. Ahmed Djamel Bouzidi Algiers University USTHB, Alger. Dipak Kumar Bhunia Autonomous University of Barcelona. Gary Birkenmeier University of Louisiana at Lafayette. Sinem Benli Izmir Institute of Technology. Delphine Boucher IRMAR, Université de Rennes 1. Engin Büyükasik Izmir Institute of Technology. xi
- 17. xii LIST OF PARTICIPANTS Victor Camillo University of Iowa. Ahmed Cherchem Algiers University USTHB, Alger. Manuel Cortés-Izurdiaga University of Málaga. Soumitra Das KPR Institute of Engineering and Technology, Coimbatore. Avanish Kumar Chaturvedi University of Allahabad, Prayagraj. Wagner Cortes Institute of Mathematics-UFRGS, Porto Alegre. Najmeh Dehghani Persian Gulf University, Bushehr. Hai Dinh Kent State University. Steven T. Dougherty University of Scranton. Müge Diril Izmir Institute of Technology. Nguyen Viet Dung Ohio University. Susan El-Deken Helwan University. Fatma Ebrahim Al-Azhar University, Cairo. Lulwah M. Al-Essa Imam Abdulrahman Bin Faisal University. Gabriella d’Este University of Milano. Sergio Estrada Universidad de Murcia. Alberto Facchini Università di Padova. Walter Ferrer Universidad de la República. Cristina Fernandez-Cordoba Autonomous University of Barcelona. Surender Jain Ohio University, Athens. Àngel Garcı́a Blàzquez University of Murcia. Diego Garcı́a Lucas Affiliation: Universidad de Murcia. Deniz Gokalp Istanbul University. Franco Guerriero Ohio University. Pedro Antonio Guil Asensio University of Murcia. Ashok Ji Gupta Indian Institue of technology(BHU), Varanasi. Fatmanur Gürsoy Yildiz Technical University. Erik Hieta-aho Aalto University. Malgorzata Hryniewicka Bialystok University. Berke Kalebogaz Hacettepe University. Yeliz Kara, Bursa Uluda University (Turkey). Fatma, Kaynarca, Afyon Kocatepe University, Afyonkarahisar. Tamer Kosan Gazi University. Jan Krempa Institute of Mathematics, Warsaw University. Arda Kor Gebze Technical University. Adrian Korban University of Chester.
- 18. LIST OF PARTICIPANTS xiii Ahmed Laghribi Université d’Artois. Ganygong Lee Chungnam National University. André Leroy Université d’Artois. Yuanlin Li Brock University. Sergio Lopez Permouth Ohio University, Athens. Najib Mahdou University S.M. Ben Abdellah, FEZ. Omar Al-mallah Al-Balqa Applied University Al-Salt. Umberto Martines Penas University of Toronto. Xavier Mary Université Paris Nanterre. Jerzy Matczuk Institute of Mathematics, Warsaw University. Sanjeev Kumar Maurya University of Allahabad. Melaibari University of Jeddah. M.Kamal Mkhallalati University of Istanbul. Intan Muchtadi-Alamsyah Institut Teknologi Bandung. Ghulam Muhiuddin University of Tabuk. Diksha Mukhija Université d’Artois. Najat Muthana University of Jeddah. Figen TAKIL MUTLU Eskisehir Technical University, Tepebasi. Mehrdad Nasernejad Institute for research in Fundamental sciences(IPM). Zahra Nazemian, University of Graz (Austria). Pace P. Nielsen Brigham Young University. Marta Nowakowska University of Silesia in Katowice. Ayse Cigdem Ozcan Hacettepe University, Ankara. Nil Orhan Ertas Bursa Technical University, Bursa. Salahattin Özdemir Dokuz Eylül University,. Jae Keol Park Busan National University. Cesar Polcino Milies University of Sao Paulo. Asiyeh Rafieipour Ohio University. Abeer Ali Rajaallah Alharbi King Abdulaziz University Branch Rabigh. Cosmin Roman Ohio State University. Louis Rowen Bar-Ilan University. Ángel del Rı́o Universidad de Murcia (Spain). Tariq Rizvi, Ohio Sate University at Lima (USA). Cosmin Roman The Ohio State University at Lima (USA). Serap Sahinkaya Tarsus University. Martha Lizbeth Sandoval Miranda Universidad Autónoma Metropolitana, Mexico City.
- 19. xiv LIST OF PARTICIPANTS Bülent Saraç Hacettepe University (Turkey). Surjeet Singh Chandigarh University. Virgilio P. Sison University of the Philippines Los Baños. Daniel Smertnig University of Graz. Agata Smoktunowicz University of Edinburgh. Anuradha Sharma IIIT-Delhi, New Delhi. Patrick Solé CNRS I2M Marseille. Amin Soofiani University of British Columbia. Ashish K. Srivastava Saint Louis University. Steve Szabo Eastern Kentucky University. Sandeep Sharma IIIT-Delhi, New Delhi. Zubeyir Turkoglu Dokuz Eyul University, Izmir Sarra Talbi U.S.T.H.B., Algiers. Melis Tekin Akcin Hacettepe University. Sultan Eylem Toksoy Hacettepe University. Yaser Tolooei Razi University. Blas Torrecillas Universidad de Almerı́a. Derya Keskin Tutuncu Hacettepe University. Gülsen Ulucak Gebze Technical University. Vaishali Varshney department of Mathematics, AMU. Maximiliano Vides Universidad Nacional delLitoral. Daniel Windisch Graz University of Technology. Jay Wood University West Michigan. Monika Yadav IIIT-Delhi, New Delhi. Tulay Yildirim Karabuk University. Eda Yildiz Yildiz Technical University, Istanbul, (Turkey). Mohamed F. Yousif The Ohio State University. Yiqiang Zhou Memorial University of Newfoundland.
- 20. Contemporary Mathematics Volume 785, 2023 https://doi.org/10.1090/conm/785/15770 A note on checkable codes over Frobenius and quasi-Frobenius rings R. R. de Araujo, A. L. M. Pereira, and C. Polcino Milies Abstract. We study checkable codes in finite Frobenius and quasi-Frobenius rings and show that the Jacobson radical of a quasi-Frobenius ring is checkable if and only if the ring is Frobenius. Also, we show that in a quasi-Frobenius ring R every maximal left ideal is checkable and that R is code-checkable if and only if R is a principal right ideal ring. Dedicated to Professor Syed Tariq Rizvi for his contribution to Ring and Mod- ule Theory. 1. Introduction Cyclic codes over a finite field F were introduced by Prange [14] in 1957 and can be realized either as ideals in a factor ring F[X]/Xn − 1 or in its isomorphic image FCn, where Cn denotes a cyclic group of order n. Extending this idea, S.D. Berman [2] in 1967 and F.J. MacWilliams [13] in 1970, independently, introduced the notion of a group code, which can be defined as a left (or right) ideal in a finite group algebra. In 2008 J.A. Wood [15] showed that code equivalence characterizes finite Frobe- nius rings. More precisely, MacWilliams [11, 12] proved that linear codes over a finite field F have the following extension property: if f : C1 → C2 is a linear iso- morphism between linear codes C1, C2 in Fn and f preserves the Hamming weight, then f extends to a monomial transformation of Fn . It is shown in [15] that if the extension property holds for linear codes over a finite ring R, then R is a finite Frobenius ring. The concept of checkable codes, due to P. Hurley and T. Hurley [7], is related to the notion zero-divisors in group rings . If RG is a finite group ring, a code (left ideal) C of RG is said to be checkable if there exists v ∈ RG such that C = {y ∈ RG : yv = 0}, i.e. if its left annihilator is a principal ideal. A group ring RG is called code-checkable if every code in RG is checkable. Jitman et al. [8, Proposition 3.1] showed that if F is a finite field and G is a finite abelian group, then the group algebra FG is code-checkable if and only if FG is a 2020 Mathematics Subject Classification. Primary 11T71, 16S34, 20C05. Secondary 16L60, 94A24. Key words and phrases. codes, checkable code, Frobenius ring, quasi-Frobenius ring. The authors were partially supported by FAPESP, Proc 2015/09162-9. c 2023 American Mathematical Society 1
- 21. 2 R. R. DE ARAUJO, A. L. M. PEREIRA, AND C. POLCINO MILIES principal ideal ring. In [1, Proposition 4.1], Abdelghany and Megahed extended this result to the case when coefficients are taken in a finite commutative ring. Recently, Borello, de la Cruz and Willems showed that the same is true for Frobenius algebras: [3, Corollary 2.3] In a Frobenius algebra A, a right (resp. left) ideal I is check- able if and only if Annl(I) resp. Annr(I) is a principal left (resp. right) ideal. As pointed out in the same paper, there are many examples of checkable codes. [3, Examples 2.4] (a) Let e = e2 be an idempotent element in A. Then the ideal eA is checkable. (b) If A is a semisimple algebra, then all right and left ideals are generated by idempotents. Thus all right and left ideals are checkable. (c) All cyclic codes are checkable, since the check equation is given by the check polynomial. (d) LCD group codes C (that is, codes for which C ∩C⊥ = {0}) are checkable since C = eKG with a self-adjoint idempotent e by [4]. The “double annihilator property” plays an important role in [3]. Actually, this property characterizes finite quasi-Frobenius rings [10]. Following this same approach, we study checkable codes also over quasi-Frobenius rings. 2. Finite Frobenius and quasi-Frobenius rings In what follows we consider Frobenius rings and quasi-Frobenius rings only in the case of finite rings (rings are always associative with identity 1 = 0). A general definition (including the infinite case) can be found in [10, Chapter 6]. For every left ideal Il R, the right ideal Annr(I) := {x ∈ R : Ix = 0} denotes the right annihilator of I. Similarly, the left ideal Annl(I) := {x ∈ R : xI = 0} is the left annihilator of a right ideal I r R. Definition 2.1. A finite ring R is a quasi-Frobenius ring (QF ring) if it satisfies the two double annihilator properties: (i) Annr(Annl(I)) = I, for all right ideal I r R; (1) (ii) Annl(Annr(I)) = I, for all left ideal I l R. (2) The above definition indicates that, in QF rings, the maps I r R → Annl(I) l R and I l R → Annr(I) r R define mutually inverse lattice anti-isomorphisms [10, Corollary 15.6]. Also, denoting by J(R) the Jacob- son radical of R and by Soc(RR) and Soc(RR) the socles of the R-modules RR and RR, i.e. the sum of all minimal left and of all minimal right ideals of R respectively, in [10, Corollary 15.7] it is shown that, in QF rings, (3) Annl(J(R)) = Soc(RR) = Soc(RR) = Annr(J(R)). If G is a finite group and R a finite QF ring, then the group ring RG is a QF ring. A direct sum of rings R = n i Ri is a QF if and only if each Ri is a QF ring. In turn, if F is a finite field, R = F F O F is an example of a ring that is not quasi-Frobenius [10]. A special class of QF rings are Frobenius rings which, in the finite case, can be defined as follows:
- 22. A NOTE ON CHECKABLE CODES OVER FROBENIUS AND QUASI-FROBENIUS RINGS 3 Definition 2.2. A finite ring R is said to be a Frobenius ring if Soc(RR) is isomorphic to (R/J(R))R and Soc(RR) is isomorphic to R(R/J(R)). Theorem 16.14 of [10] guarantees that Frobenius rings are QF rings. Honold, [6], pointed out important facts about finite Frobenius rings. In particular, in [6, Theorem 2] it is shown that only one of the isomorphisms in Definition 2.2 is needed to ensure that a finite ring R is Frobenius: Theorem 2.1. [6, Theorems 1 and 2] Let R be a finite ring. Then, following properties are equivalent: (i) R is Frobenius; (ii) Soc(RR) is cyclic; (iii) Soc(RR) is cyclic; (iv) |I| · |Annr(I)| = |R|, for every left ideal I l R; (v) |I| · |Annl(I)| = |R|, for every right ideal I r R. Some examples of Frobenius rings are semisimple rings and finite chain rings. A direct sum R = n i Ri is Frobenius if and only if every Ri is Frobenius. If R is a Frobenius ring and G is a finite group, then the matrix ring Mn(R) and the group ring RG are Frobenius rings [10]. Examples of QF ring which are not Frobenius can be found in [10, Examples 16 and 19]. 3. Checkable codes over finite QF rings Let R be a finite ring. A set C ⊂ R is a left (right) code in R if C is a left (right) ideal of R. In what follows, we always consider left ideals and hence will omit the word “left” in the sequel. Similar results hold for right ideals. The following result gives a characterization of checkable codes in finite QF rings: Theorem 3.1. Let R be a finite QF ring. A code C in R is checkable if and only if Annr(C) is a principal right ideal of R. Proof. A linear code C in R is checkable if and only if there exists v ∈ R such that C = Annl(vR). Since R is a QF ring, this is equivalent to the existence of some element v ∈ R such that Annr(C) = vR, which is a principal right ideal. Since the annihilator of the socle of a QF ring is its Jacobson radical, we have the following: Theorem 3.2. Let R be a finite QF ring. Then: (i) Soc(RR) is checkable if and only if J(R) is a principal right ideal of R. (ii) If R is local and J(R) is a principal right ideal of R, then there exists only one minimal left ideal I of R such that I is checkable. Proof. The left ideal Soc(RR) is checkable iff there exists v ∈ R such that Soc(RR) = Annl(vR). Since R is a QF ring, by (3), this is equivalent to vR = Annr(Soc(RR)) = J(R), which proves (i). To show (ii), suppose that R is local and J(R) is a principal right ideal of R. If I is a minimal left ideal of R, then I ⊂ Soc(RR), which implies J(R) = Annr(Soc(RR)) ⊂ Annr(I). Since R is local, J(R) is maximal and thus J(R) = Annr(I). So, Annr(I) is principal. By Theorem 3.1, I is checkable. Also, I =
- 23. 4 R. R. DE ARAUJO, A. L. M. PEREIRA, AND C. POLCINO MILIES Annl(J(R)) = Soc(RR), which implies that I is the unique minimal left ideal of R. 4. Code-checkable QF rings In this section, we characterize code-checkable finite QF rings. Also, we prove that every code-checkable finite QF ring is a Frobenius ring. We begin with an elementary result. Theorem 4.1. Let R be Frobenius ring R of cardinality pn such that [R : I] = p and I an ideal such that [R : I] = p, where p is a prime number, n ≥ 2. Then I is checkable. In particular, if n = 2, the ring R is code-checkable. Proof. Let I be a left ideal of R such that [R : I] = p. Then |I| = |R|/[R : I] = pn−1 . Since R is a Frobenius ring, it follows from Theorem 2.1 that |I|·|Annr(I)| = |R| = pn . So, |Annr(I)| = p. This means that Annr(I) is a principal right ideal in R, implying that I is checkable. Now, suppose that n = 2. Let I be a non-trivial left ideal of R (the cases I = 0 and I = R are trivial). Then |I| divides |R| = p2 and cannot be 1 or p2 , which means that |I| = p. So, [R : I] = |R|/|I| = p. It follows from the arguments above that I is checkable. Therefore, R is a code-checkable ring. Our next result shows a necessary and sufficient condition for a finite QF ring to be a Frobenius ring considering checkability of its Jacobson radical. Theorem 4.2. Let R be a finite QF ring. The Jacobson radical J(R) is check- able if and only if R is a Frobenius ring. Proof. If J(R) is checkable, it follows from Theorem 3.1 that Annr(J(R)) is a principal right ideal of R. Since R is a QF ring, it follows from (3) that Soc(RR) = Annr(J(R)). So, Soc(RR) is a principal right ideal in R, that is, Soc(RR) is cyclic in the right R-module RR. By theorem 2.1, this implies that R is Frobenius. Conversely, if R is a Frobenius ring, then R is a QF ring and Soc(RR) = Annr(J(R)) is cyclic, that is, Annr(J(R)) is a principal right ideal in R. By Theorem 3.1, J(R) is checkable. As an immediate consequence of Theorem 4.2 it follows that: Corollary 4.1. If a finite ring R is a QF ring and code-checkable, then R is a Frobenius ring. The next result extends [3, Corollary 2.11] to finite QF rings: Theorem 4.3. Let R be a finite QF ring. Then every maximal left ideal of R is checkable. Proof. For an R-module A, denote by l(A) the composition length of A (that is, the number of nonzero submodules in a composition series for A). Let M be a maximal left ideal of R and l = l(R). Then l(M) = l−1. Denote by M∗ the right R- module HomR(M, R) (the dual module of M) given by the action ϕ(m)r = ϕ(rm), for all ϕ ∈ M∗ , r ∈ R and m ∈ M. It follows from duality properties (e.g., see [10, Equation 15.14]) that R/Annr(M) is isomorphic to M∗ . Since l(M∗ ) = l(M), then l(Annr(M)) = 1, which means that Annr(M) is a minimal right ideal of R. Therefore, Annr(M) is a right principal ideal of R, since every minimal ideal is principal. It follows from Theorem 3.1 that M is checkable.
- 24. A NOTE ON CHECKABLE CODES OVER FROBENIUS AND QUASI-FROBENIUS RINGS 5 The following theorem shows that, for QF rings, the class of code-checkable rings is precisely the class of principal right ideal rings. In particular, this means that a commutative QF ring is code-checkable iff it is a principal ideal ring extending [3, Theorem 3.1] and [1, Theorem 4.5]. Theorem 4.4. Let R be a finite QF ring. Then, R is code-checkable if and only if R is a principal right ideal ring. Proof. Suppose that R is a code-checkable ring. Let I be a right ideal in R. Since Annl(I) is a linear code (left ideal) and R is a QF ring, Theorem 3.1 implies that I = Annr(Annl(I)) is principal. Therefore, R is a principal right ideal ring. Conversely, let C be a linear code of R. As R is a principal right ideal ring, Annr(C) is principal. It follows from Theorem 3.1 that C is checkable. Therefore, R is code-checkable. Corollary 4.2. Consider the direct sum R = n i=1 Ri of finite commutative QF rings. Then, R is code-checkable if and only if Ri is code-checkable, for all i, 1 ≤ i ≤ n. Proof. Since each Ri is a QF ring, for all 1 ≤ i ≤ n, R is also a finite QF ring. From Theorem 4.4, R is a code-checkable ring iff R is a principal ideal ring. Also, since R is commutative, R is a principal ideal ring iff every Ri is a principal ideal ring, that is, every Ri is code-checkable and the result follows. Corollary 4.3. Let R be a finite, commutative and code-checkable QF ring. Let C be a code in R. If C = Annr(C) (= Annl(C)), then C is a nilpotent ideal of R. Proof. By Theorem 4.4, R is a principal ideal ring. Then, for each code C in R, there exists v ∈ R such that Annr(C) = vR = Rv, which implies Cv = 0. If C = Annr(C), then C = Rv and v2 = 0. Therefore, C is nilpotent. References [1] Noha Abdelghany and Nefertiti Megahed, Code-checkable group rings, J. Algebra Comb. Dis- crete Struct. Appl. 4 (2017), no. 2, 115–122, DOI 10.13069/jacodesmath.284939. MR3601343 [2] S.D. Berman, On the theory of group codes, Kibernetika,1 (1967), 31-39. [3] Martino Borello, Javier De La Cruz, and Wolfgang Willems, On checkable codes in group algebras, J. Algebra Appl. 21 (2022), no. 6, Paper No. 2250125, 11, DOI 10.1142/S0219498822501250. MR4429010 [4] Javier de la Cruz and Wolfgang Willems, On group codes with complementary duals, Des. Codes Cryptogr. 86 (2018), no. 9, 2065–2073, DOI 10.1007/s10623-017-0437-2. MR3816214 [5] Marcus Greferath, Alexandr Nechaev, and Robert Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (2004), no. 3, 247–272, DOI 10.1142/S0219498804000873. MR2096449 [6] Thomas Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406–415, DOI 10.1007/PL00000451. MR1831096 [7] P. Hurley and T. Hurley, Module codes in group rings. Proc. Int. Symp. Information Theory (ISIT), 2007, pp. 1981-1985. [8] S. Jitman; S. Ling; H. Liu; X. Xie. Checkable Codes from Group Rings, arXiv:1012.5498, 2010. [9] Michael Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4 (German), Arch. Math. (Basel) 53 (1989), no. 2, 201–207, DOI 10.1007/BF01198572. MR1004279 [10] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999, DOI 10.1007/978-1-4612-0525-8. MR1653294
- 25. 6 R. R. DE ARAUJO, A. L. M. PEREIRA, AND C. POLCINO MILIES [11] Jessie MacWilliams, Error-correcting codes for multiple-level transmission, Bell System Tech. J. 40 (1961), 281–308, DOI 10.1002/j.1538-7305.1961.tb03986.x. MR141541 [12] Florence Jessie MacWilliams, Combinatorial Problems of Elementary Abelian Groups, Pro- Quest LLC, Ann Arbor, MI, 1962. Thesis (Ph.D.)–Radcliffe College. MR2939359 [13] F. J. MacWilliams, Binary codes which are ideals in the group algebra of an Abelian group, Bell System Tech. J. 49 (1970), 987–1011, DOI 10.1002/j.1538-7305.1970.tb01812.x. MR265054 [14] E. Prange, Cyclic error-correcting codes in two symbols, AFCRC-TN-57-103, USAF, Cam- bridge Research Laboratories, New York (1957) [15] Jay A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008), no. 2, 699–706, DOI 10.1090/S0002-9939-07-09164-2. MR2358511 Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, CEP 11533-160, Brazil. Email address: robson.ricardo@ifsp.edu.br Demat, Universidade Federal Rural do Rio de Janeiro, Soropedica, RJ, Brazil, CEP Seropédica - RJ, 23890-000 Email address: almp1980@ufrrj.br Instituto de Matemática e Estatı́stica, Universidade de São Paulo, Caixa Postal 66281, CEP-05315-970, Brazil. Email address: polcinomilies@gmail.com
- 26. Contemporary Mathematics Volume 785, 2023 https://doi.org/10.1090/conm/785/15771 Good classical and quantum codes from multi-twisted codes Nuh Aydin, Thomas Guidotti, and Peihan Liu Dedicated to Professor Syed Tariq Rizvi for his contribution to Ring and Module Theory Abstract. Multi-twisted (MT) codes were introduced as a generalization of quasi-twisted (QT) codes. QT codes have been known to contain many good codes. In this work, we show that codes with good parameters and desir- able properties can be obtained from MT codes. These include best known and optimal classical codes with additional properties such as reversibility and self-duality, and new and best known non-binary quantum codes obtained from special cases of MT codes. Often times best known quantum codes in the literature are obtained indirectly by considering extension rings. Our con- structions have the advantage that we obtain these codes by more direct and simpler methods. Additionally, we found theoretical results about binomials over finite fields that are useful in our search. 1. Introduction and Motivation A liner code C of length n over Fq, the finite field of order q (also denoted by GF(q)), is a vector subspace of Fn q . The three important parameters of a linear code are: the length (n), the dimension (k), and the minimum distance (d). If a code C over Fq is of length n, dimension k, and minimum distance d, then we refer to it as an [n, k, d]q-code. One of the most important and challenging problems in coding theory is to determine the optimal parameters of a linear code and to explicitly construct codes that attain them. For example, given the alphabet size q, length n, and dimension k, researchers try to determine the highest value of the minimum distance d, and also to give an explicit construction of such a code. Despite much work on this question, it still has many open cases. There are databases that contain information about best known linear codes (BKLC) and their constructions. Two of the well known databases are the maintained by M. Grassl ( [21], codetables.de) and the database contained in the Magma software [25]. Given q, n and k, there are theoretical upper bounds on d that are given in the tables mentioned above. However, it is not guaranteed and it is not always possible that they can actually be attained. This is a challenging problem for two main reasons: firstly, computing the minimum distance of a linear code is an 2020 Mathematics Subject Classification. Primary 94B05, 94B60. Key words and phrases. Multi-twisted Codes, best known linear codes, quantum codes, re- versible codes, self-dual codes. The authors were supported in part by a Kenyon Summer Science Scholars program. c 2023 American Mathematical Society 7
- 27. 8 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU NP-hard problem [32] and secondly, the number (qn − 1)(qn − q) · · · (qn − qk−1 ) (qk − 1)(qk − q) · · · (qk − qk−1) of linear codes over Fq of length n and dimension k grows fast when n, k and q increase. Therefore, it is impossible to do an exhaustive search except for very small parameters. Hence, researchers focus on sub-classes of linear codes with special algebraic properties. One very promising class has been the class of quasi-twisted (QT) codes which contains the important classes such as cyclic codes, constacyclic codes, and quasi-cyclic (QC) codes as special cases. Hundreds of BKLCs have been obtained, usually by computer searches, from the class of QT codes. More recently, a generalization of QT codes called multi-twisted (MT) codes was introduced in [4]. Therefore, MT codes are a generalization of cyclic, negacyclic, constacyclic (CC), quasi-cyclic (QC), generalized quasi-cyclic (GQC) and quasi- twisted codes (QT). Compared to QT codes, the search space for MT codes is much larger. While this increases the chances of finding new codes, it also makes the search process computationally more expensive. In this work, we focus on a special type of MT codes whose generator matrix is of the form G = [Ik | A] where Ik is the k × k identity matrix and A is a k × (n − k) circulant matrix. The idea of investigating these codes was first introduced in [10]. In addition to trying to find new linear codes from this type of MT codes, we also worked on obtaining codes that are reversible, self-dual, or self-orthogonal. Finally, we also worked on obtaining quantum codes from classical codes, specif- ically from constacyclic, QC, QT, and MT codes. The idea of quantum codes was first introduced in [28]. Later, methods of constructing quantum codes from clas- sical codes were introduced in [15] (known as the CSS construction) and [30] (the Steane construction). Since then, researchers have been working on various ways of obtaining quantum codes from classical codes. More recently, codes over extension rings have attracted a lot of attention. Many quantum codes in the literature have been obtained via the method of considering codes over an extension ring S of a base/ground ring (or field) R and using a map to eventually obtain codes over the ground ring R. We showed that in many cases, there is no need to obtain these codes in such an indirect way. We have obtained many of these best known codes presented in the literature, and even better ones, by using a direct approach. It is preferable to obtain codes in more direct ways with convenient algebraic structures. 2. Preliminaries Definition 2.1. [4] For each i = 1,. . . , , let mi be a positive integer and ai ∈ F∗ q = Fq {0}. A multi-twisted (MT) module V is an Fq[x]-module of the form V = i=1 Fq[x]/xmi − ai. A MT code is an Fq[x]-submodule of a MT module V . Equivalently, we can define a MT code in terms of the shift of a codeword. Namely, a linear code C is MT if for any codeword c = (c1,0, . . . , c1,m1−1; c2,0, . . . , c2,m2−1; . . . ; c,0, . . . , c,m−1) ∈ C, its multi-twisted shift (a1c1,m1−1, c1,0, . . . , c1,m1−2; a2c2,m2−1, c2,0, . . . , c2,m2−2; . . . ; ac,m−1, . . . , c,m−2)
- 28. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 9 is also a codeword. If we identify a vector c with C(x) = (c1(x), c2(x), . . . , c(x)) where ci(x) = ci,0 + ci,1x + · · · + ci,mi−1xmi−1 , then the MT shift corresponds to xC(x) = (xc1(x) mod xm1 − a1, . . . , xc(x) mod xm − a). The following are some of the most important special cases of MT codes: • a1 = a2 = · · · = a gives QT codes • a1 = a2 = · · · = a = 1 gives QC codes • = 1, a1 = 1 gives cyclic codes • = 1, a1 = −1 gives negacyclic codes • = 1 gives constacyclic codes There has been much research on QT codes, a special case of MT codes. Hun- dreds of new codes have been obtained from QT codes by computer searches. An algorithm called ASR ([3]) has been particularly effective in this regard. The ASR algorithm is based on the following theorem. Theorem 2.2. [3] Let C be a 1-generator QT code of length n = m over Fq with a generator of the form (g(x)f1(x), g(x)f2(x), ..., g(x)f(x)) where xm − a = g(x)h(x) and gcd(h(x), fi(x)) = 1 for all i = 1, . . . , . Then dim(C) = m − deg(g(x)), and d(C) ≥ · d where d is the minimum distance of the constacyclic code Cg generated by g(x). This algorithm has been refined and automatized in more recent works such as ([5], [6], [7], [8], [9]) and dozens of record breaking codes have been obtained over every finite field Fq, for q = 2, 3, 4, 5, 7, 8, 9 through its implementation. Moreover, it has been further generalized in [10] and more new codes were discovered that would have been missed by its earlier versions. We can consider MT analogues of the QT codes given in the previous theorem. The following theorem gives a basic fact about duals of such codes. Theorem 2.3. Let C = (g1(x), g2(x), . . . , g(x)) be an MT code such that the constacyclic codes generated by gi(x) have the same length n and the same dimension k for i = 1, 2, . . . , . Let Di be the dual of gi(x). Let PD = D1 × D2 × · · · × D. Then PD is contained in C⊥ . Proof. Let Gi be a generator matrix of Ci = gi(x), so a generator matrix G of C is given by G = G1 G2 G3 . . . G k× i=1 ni . Let Hi be a generator matrix of Di = C⊥ i . Then a generator matrix GPD of PD is given by GPD = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ H1 0 0 . . . 0 0 H2 0 . . . 0 0 0 H3 . . . 0 ... ... ... ... ... 0 0 0 . . . H ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (n−k)× i=1 ni . It is readily verified that G·GPD = 0. Note that the dimension of C⊥ is i=1 ni−k and the dimension of PD is i=1(ni − k), so the dimension of PD is less than or equal to the dimension of C⊥ . Hence, PD ⊆ C⊥ .
- 29. 10 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU 3. A Special Type of MT Codes We first introduced a special type of MT codes that we called ICY codes in [10]. The basic idea for this construction is inspired by the well known fact that any linear code is equivalent to a linear code whose generator matrix is of the form G = [Ik | A] where Ik is the k × k identity matrix and A is a k × (n − k) matrix. A code with a generator matrix in this form is called a systematic code. In this search method, we begin with a k × k identity matrix on the left and append it with k × (n − k) circulant matrices on the right. The circulant matrix is constructed using a generator polynomial g(x)|(xn−k − a) of a constacyclic code (obtained using the partition program from [9]). For each polynomial g(x), we consider a number of circulant matrices obtained from polynomials g(x)·f(x) where gcd(f(x), h(x)) = 1, with the goal of finding a code with minimum distance equal to or greater than the BKLCs for parameters [n, k]q. In [10] we were able to find many good codes over GF(2) and GF(5). In this paper we present a number of optimal codes over GF(2) with parameters [2·k, k] obtained from an implementation of this method. A number of these codes we found are also self-dual, which makes them even more useful. Another property of a code that is useful in some applications, such as DNA codes is reversibility, is reversibility. A code is reversible if, whenever (c0, c1, . . . , cn−1) is a codeword, so is (cn−1, cn−2, . . . , c0). Lemma 3.1. If f(x) is a monomial and the block length of f(x) is the same as the block length of the identituy matrix, then the MT code C over GF(2) generated by 1, f(x) is always self-dual, reversible and of minimum distance 2. In particular, if the monomial is a constant, then C is a cyclic code. Proof. Let f(x) = xi be a monomial and the block length of f(x) be the same as k × k identity matrix. Then the generator matrix of the MT code C over GF(2) generated by 1, f(x) is given by: G = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 . . . 0 0 . . . 1 0 . . . 0 1 . . . 0 0 . . . 0 1 . . . ... ... ... ... ... ... ... ... ... 0 0 . . . 1 . . . 1 0 0 . . . ⎞ ⎟ ⎟ ⎟ ⎠ k×2k . It is easy to verify that the reverse of the jt h row is the (k − i + 1 − j)t h row, so it follows that C is reversible. Also note that each row of G is orthogonal to itself and also orthogonal to every other row, so given k = 2k/2 we know C is also self-dual. Moreover, since each row is of weight 2, and both matrices are circulant, it follows that C is of minimum distance 2. Specifically, if f(x) = 1, then the generator matrix G is given by G = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 . . . 0 1 . . . 0 . . . 0 1 . . . 0 0 1 0 . . . ... ... ... ... ... ... ... ... 0 0 . . . 1 . . . 0 . . . 1 ⎞ ⎟ ⎟ ⎟ ⎠ k×2k . Hence, the code is a cyclic code of length 2k generated by f(x) = (xk − 1)|(x2k − 1). The following codes in Table 1 are obtained by the ICY method where we chose n = 2k, so each one of these codes has parameters [2k, k, d]. All of these codes are
- 30. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 11 optimal with additional properties of being reversible and/or self orthogonal. For the circulant matrix on the right, we used a random m × m circulant matrix by choosing a random vector and taking its cyclic shifts. This polynomial is given in the table. We also checked the conditions for reversibility and self-duality. Note that self-orthogonality is the same as self-duality since we set n = 2k. In this search, the binary field fields turned out to be most promising. In this and every subsequent table presented in this paper the terms of the generator polynomials are listed in ascending order from left to right. For instance, the generator polynomial [111101] of the fourth code in this table is the vector representation of g(x) = 1 + x + x2 + x3 + x5 . Table 1: Optimal codes from the ICY method with several desirable properties. Here, ∗ delineates a reversible code, and ◦ delineates a self- dual code. [n, k, d]q Polynomials [8, 4, 4]2 ∗◦ g = [111] [10, 5, 4]2 g = [10101] [12, 6, 4]2 ∗◦ g = [110111] [14, 7, 4]2 g = [111101] [20, 10, 6]2 g = [11111011] [24, 12, 8]2 ∗◦ g = [101111011] [28, 14, 8]2 g = [11101111001001] [32, 16, 8]2 ∗◦ g = [1100100001100011] [36, 18, 8]2 ∗◦ g = [111010010111001] [42, 21, 10]2 g = [11110011011011101001] [48, 24, 12]2 ∗◦ g = [10111101001011110011101] [16, 8, 6]3 ∗ g = [212222] [18, 9, 6]3 ∗ g = [201002222] [20, 10, 7]3 ∗ g = [112012102] [30, 15, 9]3 ∗ g = [201212222200121] [14, 7, 6]4 ∗◦ g = [ααα2 αα2 α2 ] [20, 10, 8]4 ∗◦ g = [αα1α00α2 1α2 α2 ] [24, 12, 9]4 g = [10αα2 αα2 α10α1] [14, 7, 6]5 g = [3343442] [16, 8, 7]5 g = [34404411] [12, 6, 6]7 g = [551641] [14, 7, 7]7 g = [5226211] [16, 8, 7]7 g = [42325344] [18, 9, 8]7 g = [524635101] The next table presents a list of optimal self-dual codes together with their weight enumerators obtained by this method. For codes over F4 = {0, 1, α, α + 1} in this paper, α is a root of x2 +x+1 over F2. Again, ∗ delineates a reversible code. The table also presents the weight enumerator of each code and for each binary code specifies whether it is Type I or Type II.
- 31. 12 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU Table 2: Optimal self-dual codes from the ICY method. [n, k, d]q Type weight enumerator Polynomials [8, 4, 4]2 ∗ II A8 + 14A4 + 1 g = [111] [12, 6, 4]2 I A12 + 15A8 + 32A6 + 15A4 + 1 g = [110111] [16, 8, 4]2 ∗ II A16 + 28A12 + 198A8 + 28A4 + 1 g = [110111] [16, 8, 4]2 ∗ I A16 +12A12 +64A10 +102A8 +64A6 + 12A4 + 1 g = [1010111] [18, 9, 4]2 ∗ I A18 + 9A14 + 75A12 + 171A10 + 171A8 + 75A6 + 9A4 + 1 g = [11011001] [20, 10, 4]2 ∗ I A20 + 45A16 + 210A12 + 512A10 + 210A8 + 45A4 + 1 g = [1011111111] [22, 11, 6]2 ∗ I A22 + 77A16 + 330A14 + 616A12 + 616A10 + 330A8 + 77A6 + 1 g = [1001011] [26, 13, 6]2 I A26 + 52A20 + 390A18 + 1313A16 + 2340A14 + 2340A12 + 1313A10 + 390A8 + 52A6 + 1 g = [10111110111] [52, 26, 10]2 I A52 +442A42 +6188A40 +53040A38 + 308958A36 + 1270360A34 + 3754569A32 + 8065616A30 + 12707500A28 + 14775516A26 + 12707500A24 + 8065616A22 + 3754569A20 + 1270360A18 + 308958A16 + 53040A14 + 6188A12 + 442A10 + 1 g = [1001111100101010100100101] [14, 7, 6]4 ∗ Euclidean 318A14 + 1302A13 + 2940A12 + 3990A11 + 3738A10 + 2226A9 + 1155A8 + 546A7 + 168A6 + 1 g = [α1αα2ααα2] [16, 8, 6]4 ∗ Euclidean 681A16 + 3696A15 + 7896A14 + 14112A13 + 16464A12 + 9408A11 + 7392A10 +4704A9 +678A8 +336A7 + 168A6 + 1 g = [αα1α2αα2] [20, 10, 8]4 ∗ Euclidean 171A12 +432A11 +864A10 +1440A9 + 459A8 + 432A7 + 288A6 + 9A4 + 1 g = [αα1α00α2α2α2] [22, 11, 8]4 ∗ Euclidean 171A12 +432A11 +864A10 +1440A9 + 459A8 + 432A7 + 288A6 + 9A4 + 1 g = [α2ααα2010αα2α2α] 4. Reversible Codes Reversible codes are essential in application of coding theory to DNA computing [27]. Each single DNA strand is composed of a sequence of four bases nucleotides (adenine (A), guanine (G), thymine (T), cytosine (C) ) and it is paired up with a complementary strand to form a double helix [1]. Finding reversible codes is an essential requirement in order to find codes suitable for DNA computing. Also, reversible codes may be used in certain data storage applications [26]. For example, if the code is reversible then one can read the stored data from either end of a block. In this paper, we present a general method to construct reversible -block MT, QT and QC codes, and a number of good reversible QC codes over GF(2) are obtained using this method. The codes presented in Table 3 are reversible and self-orthogonal QC with the same parameters as BKLCs over GF(2). Hence, they are more desirable than BKLCs that do not have these properties. In checking the reversibility of these codes, we used the conditions for reversibility of QC codes given in [31]. Theo- rem 1 in section IV of [31] presents three conditions for reversibility of QC codes with generator matrices of the form [g(x), g(x) · f(x)], where g(x)|(xm − 1) and gcd(f(x), h(x))) = 1. After finding all nonequivalent generators g(x) for a given
- 32. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 13 block length m, we check the first two conditions on g(x) and h(x) before proceed- ing. If these two checks pass, then we generate random polynomials f(x) that satisfy the third condition in the theorem, resulting in a reversible QC code. Additionally, since these QC codes are constructed over GF(2) they have the added property of being self-orthogonal. Theorem 1 also provides conditions for self-orthogonality of QC codes with such generator matrices and in the binary case these conditions are the same as those for reversibility. For a polynomial f(x), it is customary to denote its reciprocal by f(x)∗ . Theorem 4.1. Let gi|(xni − ai) and gcd(hi, fi) = 1 for i = 1, 2, . . . , , where hi = xni −ai gi . If deg(gifi) = deg(g+1−if+1−i), ni = n+1−i and gifi = (g+1−if+1−i)∗ for i = 1, 2, .., , then the MT code C generated by (g1f1, . . . , gf) is reversible. Proof. Let gifi = ai,0 + ai,1x + · · · + ai,ki xki , so a generator matrix of this MT code with blocks is given by G = G1 G2 . . . G−1 G , where Gi is Gi = ⎛ ⎜ ⎜ ⎜ ⎝ ai,0 ai,1 ai,2 . . . ai,ki 0 0 . . . 0 ai,0 ai,1 . . . ai,k−1 ai,ki 0 . . . . . . ... ... ... ... ... ... ... 0 0 . . . ai,0 ai,1 ai,2 . . . ai,ki ⎞ ⎟ ⎟ ⎟ ⎠ (ni−ki)×ni . Hence, the reversed first row of G is given by (0, . . . , 0, a,k , a,k−1, . . . , a,1, a,0; 0, . . . , 0, a−1,k−1 , a−1,k−1−1, . . . , a−1,1, a−1,0; . . . 0, . . . , 0, a1,k1 , a1,k1−1, . . . , a1,1, a1,0). Note that gifi = (g+1−if+1−i)∗ for i = 1, 2, .., , so ai,j = a+1−i,k−j+1 for i = 1, 2, .., and j = 1, 2, . . . , k. Also, since the degree of all deg(gifi) = deg(g+1−if+1−i), it is not hard to see that the reversed first row is the (n − k)th row, and in general the reversed jth row is the same as (n − k + 1 − j)th row. Therefore, the reverse of each row in G is in G, so C is reversible. As mentioned previously all of these codes are constructed using the conditions for a 2-QC code to be reversible found in [31]. This paper gives conditions on the generator polynomials g(x) and f(x) · g(x) mod (xm − 1) with gcd(h(x), f(x)) = 1 so the first polynomial presented in this table is g(x), and the second one is f(x)·g(x) mod (xm − 1).
- 33. 14 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU Table 3: Reversible and self-orthogonal binary QC codes with best known parameters [n, k, d] Polynomials [60, 28, 12] [00101100111001101100101001101], [111] [60, 29, 12] [000111000101011000100100011111], [11] [72, 29, 16] [011111101110101111101101100111101011], [10111101] [72, 30, 16] [0110111100000111111100010010111001], [1101011] [80, 37, 16] [1010011110001011110000000111110001110001], [1111] [80, 38, 16] [0011111110101010100100101101110101011], [101] [84, 24, 24] [111100010111100111000111101110000000100101], [1010100000000010101] [84, 25, 24] [001111100001011100110011110100001101000111], [110011111111110011] [84, 26, 24] [001101010011000011101100000100001110001101], [11100000000000111] [84, 27, 24] [001001101000110101101001110110000011000001], [1011111111111101] [90, 28, 24] [011111010111100100001011100011000011110000001], [110111011110111011] [90, 34, 20] [00010111011111110110000000011010111111101101], [110001100011] [96, 36, 20] [10010100100000001100000001110111001001100111001], [1010001000101] [96, 37, 20] [01100110011010101011011110010011101110010010011], [100110011001] [96, 38, 20] [110011110011101110010010111100100111011010101], [11101110111] [100, 41, 20] [00110100011011100110001010100100101010000111001001], [1111111111] [102, 34, 24] [10011111000011100010010001100001100001101110101], [101110111111011101] [102, 42, 20] [1010110010011100000100111101101100110001000000101], [1001111001] [104, 36, 24] [1010111000101110001001010010110111001001101110110101], [11110000000001111] [108, 46, 20] [110000001010000001101111000101010010011011101000001], [111111111] [110, 40, 24] [110111111111101000011111110110011000010001011111000111], [1111100000011111] 5. Quantum Codes In comparison to classical information theory, the field of quantum information theory is relatively young. The idea of quantum error correcting codes was first introduced in [29] and [14]. A method of constructing quantum error correcting codes (QECC) was given in [15]. Since then researchers have investigated various methods of using classical error correcting codes to construct new QECCs. The majority of the methods have been based on the CSS construction given in [15]. In this method, self-dual, self-orthogonal and dual-containing linear codes are used to construct quantum codes. The CSS construction requires two linear codes C1 and C2 such that C⊥ 2 ⊆ C1. Hence, if C1 is a self-dual code, then we can construct a
- 34. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 15 CSS quantum code using C1 alone since C⊥ 1 ⊆ C1. If C1 is self-orthogonal, then we can construct a CSS quantum code with C⊥ 1 and C1 since C1 ⊆ C⊥ 1 . Similarly in the case C1 is a dual-containing code. The last few tables in this section present a number of good QECCs obtained from both the CSS method and the method of additive codes. Many of these codes have the same parameters as those found in the literature, however we obtained them using more direct and simpler constructions. Additionally, we have found other codes whose parameters do not appear in the literature. The following codes are constructed using the method of quantum codes con- structed from additive codes [15]. After generating all divisors g(x) of xm − a that generate non-equivalent constacyclic codes and finding a generator matrix of the form [Cg(x)·f1(x), Cg(x)·f2(x)] where gcd(fi(x), h(x))) = 1 and Cg(x) represents the circulant matrix from g(x), we construct an additive code from this generator matrix and test it for symplectic self orthogonality. If the code passes this check, then we construct a quantum code from this additive code and compute its minimum distance. Then we can compare the minimum distances of these codes against the comparable best known QECCs given in the database codetables.de. In Table 4, all of the codes are optimal quantum QT codes constructed over GF(22 ). Note that α is a root of the irreducible polynomial x2 +x+1 over GF(2). Table 4 : Optimal quantum codes over GF(22 ) from QT Codes [n, k, d] Polynomials [[54, 47, 2]] [00α2 1110α000α2 1110α000α2 1110α], [α2 αα2 01α2 α0αα2 αα2 01α2 α0αα2 αα2 01α2 α0α] [[56, 50, 2]] [10000α2 αα1α2 1α2 α1α2 αααα100α2 1α2 10α2 ], [α2 10011ααα2 1α0α2 α2 0αα2 α2 αα110α1α2 ] [[70, 64, 2]] [10α2 α010α0αα1ααα001ααα2 10α0α1α2 00α2 110α], [α2 01α2 ααα2 αα2 101α2 α2 01α2 α2 0α2 αα1011α0α2 α2 α2 1α0α] [[88, 80, 2]] [100α2 1αα101α2 ααα2 1α01α2 α2 α2 0011α0α2 α2 010αα2 α2 α0α2 10ααα1], [0α2 11α00α2 000100αα2 α2 101α2 1α1α2 α2 0αα1αααα2 αα011α2 αα2 1α2 ] [[98, 91, 2]] [11α2 101011α2 101011α2 101011α2 101011α2 101011α2 101011α2 101], [αα000α2 0αα000α2 0αα000α2 0αα000α2 0αα000α2 0αα000α2 0αα000α2 ] We also found some QECCs whose parameters do not appear in the literature. Hence, we consider them new quantum codes. The inspiration for this idea came from [17]. These codes are obtained using direct sums of two cyclic codes, with generators g1(x), g2(x) that are divisors of xm − 1. Each gi(x) generates a dual containing cyclic code Ci = gi(x) which implies that their direct sum C1 × C2 is also dual containing. So, the CSS method can be used with the direct sum code and its dual. The resultant QECC will have parameters [[2 · m, 2 · k − 2 · m, d]]q. Here, k is defined to be the dimension of the direct sum code which is equal to k1 + k2, the sum of the dimensions of the individual cyclic codes, and d is the minimum of the minimum distances of the cyclic codes generated by g1(x) and g2(x). Table 5 contains these new codes.
- 35. 16 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU Table 5 : New QECCs [[n, k, d]]q Polynomials [[60, 54, 2]]3 [21], [201] [[72, 68, 2]]3 [21], [21] [[84, 80, 2]]3 [21], [21] [[100, 96, 2]]5 [41], [41] Next, we have a number of codes that have the same parameters as the best known QECCs presented in the literature. Our codes have the advantage that they are obtained via a more direct and simpler construction. They were all obtained using the same method as the codes in the previous table. Codes with the same parameters listed in Table 6 were found using more complex and indirect methods detailed in [11], [2], [24] , [18], [19], [13]. These constructions usually involve considering an extension ring S of a ground ring or field R, constructing a code over the extension ring, then coming back to R using some sort of Gray map. We argue that it is more desirable to construct codes with the same parameters using a more direct approach. Table 6 : QECC ties [[n, k, d]]q Polynomials Source [[72, 66, 2]]3 [21], [111] [24] [[84, 78, 2]]3 [21], [201] [11] [[40, 36, 2]5 [41], [41] [19] [[40, 34, 2]]5 [41], [131] [11] [[40, 28, 3]]5 [3011], [3011] [19] [[60, 54, 2]]5 [41], [131] [24] [[90, 84, 2]]5 [41], [131] [18] [[100, 94, 2]]5 [41], [401] [11] [[140, 134, 2]]5 [41], [401] [13] 5.1. New Quantum Codes from Constacyclic Codes and a Result About Binomials. In this section, we obtain new QECCs that are constructed by the method of CSS construction from constacyclic codes over finite fields. These codes are new in the sense that either there do not exist codes with these param- eters in the literature (to the best of our knowledge) or the minimum distances of our codes are higher than the codes that are presented in the literature. Our work on this also led to a couple of theoretical results about the binomials of the form xn − a over Fq. The first one (Lemma 2 below) is similar to Theorem 4.1 in [4]. We start with the results on binomials. We will be using the fact that the Frobenius map x → xpm on a finite field Fq of characteristic p is a permutation of Fq for any positive integer m. In particular, for any α ∈ Fq and for any m ∈ Z+ , there exists a unique β ∈ Fq such that α = βpm . Lemma 5.1. Let n ∈ Z+ , q be a prime power and a, b ∈ Fq such that a = b. Then gcd(xn − a, xn − b) = 1.
- 36. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 17 Proof. It suffices to show that the polynomials xn − a and xn − b have no common roots. Let δ be a root of xn − a. Then, δn − b = a − b = 0 It follows that gcd(xn − a, xn − b) = 1. Theorem 5.2. Let q be a power of prime p, and let g(x) = xn − a ∈ Fq[x]. Then all irreducible factors of xn − a have the same multiplicity pz n, where z is the largest positive integer such that pz divides n. Proof. Let n = pz · b, where gcd(b, p) = 1. Then xn − a can be written as xn − a = xpz ·b − a = (xb − a )pz , where a ∈ Fq. Let’s call f(x) = xb − a , and then consider gcd(f(x), f (x)). Note that b = 0 and xb − a is not a multiple of x, so it follows that gcd(f(x), f (x)) = gcd(xb − a , bxb−1 ) = 1. Hence, all irreducible factors of f(x) = xb − a have the same multiplicity 1. Therefore, all irreducible factors of g(x) = xn − a = fpz (x) have the same multiplicity pz . By the definition of CSS construction, we need two codes such that one is contained in the dual of the other one. By the well known fact about inclusion of ideals, we know that g(x)f(x) ⊆ g(x) and we consider C⊥ 2 = g(x)f(x) ⊆ g(x) = C1. Here g(x) is a divisor of a binomial xn − a and it generates a constacyclic code. Since for a given n ∈ Z+ and distinct elements α, β ∈ Fq, gcd(xn −α, xn −β) = 1, it follows that for any divisor g(x)|(xn − α), we have g(x)f(x) (xn − β) for any f(x) ∈ Fq[x]. In other words, for a given C1 = g(x), where g(x)|(xn − α), all possible constacyclic codes of the same length C⊥ 2 such that C⊥ 2 ⊆ C1 have the form C⊥ 2 = g(x)f(x), where g(x)f(x)|(xn − α). Hence, for given α ∈ Fq and n, we only need to consider gi(x)|(xn − α) and gi(x)fi(x)|(xn − α). By Lemma 5.1, any generator of C⊥ 2 will be a divisor of xn − α, but not of xn − β for β = α. Also, based on the multiplicity of all irreducible factors of xn − α, for any g(x), we can determine all g(x)f(x) by enumerating all possible combinations of multiplicities of irreducible factors. Therefore, our algorithm goes through every possible pair of g(x) and g(x)f(x), and we view each g(x) as C1 and each g(x)f(x) as C⊥ 2 in the CSS construction. This way, we obtained many new QECCs. The new codes we obtained are of two types: i) those that have higher minimum distances than the codes given in the literature, ii) those whose parameters do not appear in the literature. We present them in the following two tables. Table 7 contains references to the literature where a comparable code is given.
- 37. 18 NUH AYDIN, THOMAS GUIDOTTI, AND PEIHAN LIU Table 7 : New QECCs that have higher minimum distances than the codes given in the literature KnownQECC Ref New [[n, k, d]]q g(x) g(x)f(x) [[11, 1, 4]]5 [23] [[11, 1, 5]]5 [124114], [1122031] [[30, 20, 3]]5 [24] [[30, 20, 4]]5 [142241], [14401243413033031434210441] [[40, 32, 2]]5 [2] [[40, 32, 3]]5 [12342], [12030222342024440142411123141333 40313] [[93, 75, 2]]5 [12] [[93, 75, 4]]5 [1032243334],[1214314012224022200134214311 4334244231242301201241431231121012330223 41423121100111211] [[60, 52, 2]]5 [24] [[60, 52, 3]]5 [10434],[101321202243220133133431344440014 124413202 440312032014321] [[78, 60, 3]]3 [12] [[78, 60, 4]]5 [1120201112],[12022010222111011212012012102 20211120012212110210210000100012101020022] [[18, 2, 3]]7 [22] [[18, 2, 6]]7 [143561634], [14631236125] [[12, 4, 3]]13 [20] [[12, 4, 5]]13 [18554], [184835595] [[24, 18, 2]]17 [20] [[24, 18, 3]]17 [1332], [9511521119410532108411631] [[21, 7, 2]]19 [16] [[21, 7, 3]]19 [19999998], [11313131313131319999991] The final table shows the set of codes we have found whose parameters do not appear in the literature. Table 8 : New QECCs [[n, k, d]]q Polynomials [[61, 41, 5]]3 [10011111001], [1100101022020121010222110210221110202120101102 020022] [[52, 7, 6]]3 [12211120111], [100000202100102111] [[52, 35, 4]]3 [110000012], [11020221221200111012111020100112002220020221] [[52, 15, 5]]3 [12010202], [12200202101110221121201] [[40, 11, 6]]3 [112100201012],[11112000221100111020001] [[40, 22, 5]]3 [1222100011], [12220111102211221220021121211111] [[30, 2, 8]]3 [101101000202101],[10211201020112201] [[32, 2, 7]]3 [1211021211201002],[122010112001001202] [[30, 10, 4]]3 [120212022101],[1211111102001211202201] [[30, 16, 4]]3 [11122111], [110112012020020210211011] [[33, 1, 10]]3 [12000021012021021],[111000222111222222] [[35, 4, 6]]3 [12122221121022], [101020002212012112] [[36, 1, 8]]3 [11120020102002111],[100110211221022002] [[36, 2, 8]]3 [12212100100121221],[1200122110112210021] [[36, 18, 4]]3 [1110220111], [1102010202122112212020102011] [[20, 4, 5]]3 [112212211], [1122201022211] [[23, 1, 8]]3 [100202011222],[1202121101001] [[26, 9, 6]]3 [111122121], [112200100222020102] [[28, 0, 9]]3 [122221000112122],[122221000112122] [[28, 12, 6]]3 [111000121], [121100111020121002111] [[14, 2, 5]]3 [1101011], [111202111] [[11, 1, 5]]5 [124114], [1122031]
- 38. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 19 [[12, 4, 4]]5 [14102], [140124402] [[16, 1, 6]]5 [1320214], [11112222] [[20, 4, 6]]5 [121201212], [1231323013112] [[30, 20, 4]]5 [142241], [14401243413033031434210441] [[10, 2, 4]]7 [12134], [1230661] [[14, 6, 4]]7 [12056], [12462526421] [[40, 28, 4]]7 [1455541], [11152236066630031463035562341504546] [[12, 2, 5]]13 [154816], [1582412411] 5.2. New Quantum Codes from Constacyclic Random Search. In this section, we introduce another method to obtain quantum codes from classical codes that we call constacyclic random (CR) search. We have obtained several new quan- tum codes (codes with higher minimum distances than the ones in the literature) from this search. There is a small difference between the CR search and the earlier method of constacyclic quantum search. In the CR search, we first find all g(x)|(xn − a), then find a large number of random polynomials f(x), and construct CSS codes with g(x) and g(x)f(x). By this method, we obtained a new [[11, 1, 5]]5-code which was also found by the earlier search method and listed in Table 7. Additionally, we found a [[19, 1, 9]]17- code from the generators g = x9 +4x8 +15x7 +15x6 +6x5 +x4 +12x3 +2x2 +3x+16 and gf = x10 +3x9 +11x8 +8x6 +12x5 +11x4 +7x3 +x2 +13x+1, which improves the minimum distance of the known QECC [[19, 1, 5]]17-code in [23] by 4 units. Finally, we have found a [[30, 10, 3]]19-code from the generators g = x4 +11x3 +14x2 +3x+18 and gf = x14 + 11x12 + 6x11 + 8x10 + x9 + 4x8 + 17x7 + 4x6 + x5 + 13x4 + 7x3 + 17x2 + 15x + 8, which improves the minimum distance of the known QECC code with parameters [[30, 10, 2]]19 given in [16] by 1 unit. Our computational results suggest that this is a promising search method that might yield even more new codes. Acknowledgments This work was supported by Kenyon College Summer Science Scholars program. References [1] Taher Abualrub, Ali Ghrayeb, and Xiang Nian Zeng, Construction of cyclic codes over GF(4) for DNA computing, J. Franklin Inst. 343 (2006), no. 4-5, 448–457, DOI 10.1016/j.jfranklin.2006.02.009. MR2262559 [2] Mohammad Ashraf and Ghulam Mohammad, Quantum codes from cyclic codes over Fq + uFq +vFq +uvFq, Quantum Inf. Process. 15 (2016), no. 10, 4089–4098, DOI 10.1007/s11128- 016-1379-8. MR3547872 [3] Nuh Aydin, Irfan Siap, and Dijen K. Ray-Chaudhuri, The structure of 1-generator quasi- twisted codes and new linear codes, Des. Codes Cryptogr. 24 (2001), no. 3, 313–326, DOI 10.1023/A:1011283523000. MR1857145 [4] Nuh Aydin and Ajdin Halilović, A generalization of quasi-twisted codes: multi-twisted codes, Finite Fields Appl. 45 (2017), 96–106, DOI 10.1016/j.ffa.2016.12.002. MR3631356 [5] Nuh Aydin, Nicholas Connolly, and John Murphree, New binary linear codes from quasi-cyclic codes and an augmentation algorithm, Appl. Algebra Engrg. Comm. Comput. 28 (2017), no. 4, 339–350, DOI 10.1007/s00200-017-0327-x. MR3679335 [6] Nuh Aydin, Nicholas Connolly, and Markus Grassl, Some results on the structure of consta- cyclic codes and new linear codes over GF(7) from quasi-twisted codes, Adv. Math. Commun. 11 (2017), no. 1, 245–258, DOI 10.3934/amc.2017016. MR3613297
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- 40. GOOD CLASSICAL AND QUANTUM CODES FROM MULTI-TWISTED CODES 21 [30] Andrew M. Steane, Enlargement of Calderbank-Shor-Steane quantum codes, IEEE Trans. Inform. Theory 45 (1999), no. 7, 2492–2495, DOI 10.1109/18.796388. MR1725135 [31] R. Takieldin, H. Matsui, On reversibility and self-duality for some classes of quasi-cyclic codes, IEEE Access. 8 (2020), 143285–143293. [32] Alexander Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43 (1997), no. 6, 1757–1766, DOI 10.1109/18.641542. MR1481035 Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022 Email address: aydinn@kenyon.edu Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022 Email address: guidotti1@kenyon.edu Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022 Current address: Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 48104 Email address: paulliu@umich.edu
- 42. Contemporary Mathematics Volume 785, 2023 https://doi.org/10.1090/conm/785/15772 Evaluation of iterated Ore polynomials and skew Reed-Muller codes Nabil Bennenni and André Leroy Dedicated to Professor Syed Tariq Rizvi for his contribution to Ring and Module Theory. Abstract. In this paper, we study two ways of evaluating iterated Ore poly- nomials. We provide many examples and compare these evaluations. We use the evaluation maps to construct Reed-Muller codes and compute explicitly some of the data that are associated to such codes. Introduction The evaluation of polynomials is a keystone in many branches of mathematics. In noncommutative algebra one of the most attractive notion of polynomials is the Ore polynomials (a.k.a. skew polynomials). There are plenty of works dealing with the evaluations of these polynomials with coefficients in division rings or prime rings [7],[8]. In coding theory the evaluation of polynomials is used, in particular, for constructing Reed-Muller codes [10], [5], [4]. Let us recall the general definition of the skew polynomial ring. Let K be a ring, σ ∈ End(K), and δ a skew σ-derivation on K (i.e. δ is an additive map from K to K such that for any a, b ∈ K, δ(ab) = σ(a)δ(b)+δ(a)b). The elements of a skew polyno- mial ring R = K[t; σ, δ] are polynomials n i=0 aiti with coefficients ai ∈ K written on the left. The addition of these polynomials is the usual one and the multiplica- tion is based on the commutation rule ta = σ(a)t+δ(a), for any a ∈ K. With these two operations, R becomes a ring. When K is a division ring, R is a left principal ideal domain. If σ is an automorphism of the division ring K then R is also a right principal ideal domain. This construction can be iterated: we consider a ring K, a sequence of skew polynomial rings R1 = K[t1; σ1, δ1], where σ1 ∈ End(K) and δ1 is a σ1-derivation of K. We then build R2 = R1[t1; σ2, δ2] = K[t1; σ1, δ1][t2; σ2, δ2], where σ2 ∈ End(R1) and δ2 is a σ2-derivation of R1. Continuing this construc- tion we arrived at the iterated polynomial ring in n noncommutative variables R = Rn = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn]. The evaluation of skew polyno- mials has a very different behavior than the usual one for commutative variables. Most of the literature is related to right evaluation of univariate (i.e. n = 1) polynomials with coefficients in a division ring (e.g. [8], [7]). For the case when the base ring K is not necessarily a division ring, the pseudo-linear transforma- tions are very useful [7]. When K is a division ring the left evaluation was 2020 Mathematics Subject Classification. Primary 94B05, 11T71, 14G50; Secondary 16L30. Key words and phrases. Evaluation of iterated Ore polynomials, skew Reed-Muller codes. c 2023 American Mathematical Society 23
- 43. 24 NABIL BENNENNI AND ANDRÉ LEROY also considered even when σ is not onto [9]. While considering the evaluation of a polynomial f(t1, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] at a point (a1, . . . , an) ∈ Kn , we are quickly facing a “noncommutative phenomenon”: the ideal I = R(t1 − a1) + · · · + R(t2 − a2) can be equal to R. We will show how it is possible to avoid this problem. Reed-Muller codes over F2 were introduced by Reed in 1954 [10]. Kasami, Lin and Peterson in [6] extended the definition to any finite field Fq. Reed-Muller codes are very useful in cryptography. Indeed, they show good resistance to linear cryptanalysis. The interested reader may consult [4], for more details. Reed-Muller codes are amongst the simplest examples of the class of geometrical codes, which also includes Euclidean geometry and projective geometry codes. The paper is organized as follows. Section 2 presents two ways of evaluating iterated polynomials in a general context. We provide many examples and compare these evaluations. In Section 3, we apply the evaluation maps to construct Reed- Muller codes and compute some of the data that are associated to such codes. 1. Polynomial maps in n variables In this section, we define a notion of evaluation for polynomials in an iterated Ore extension R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] over an arbitrary ring K (with unity). Throughout the paper we require that σi(K) ⊂ K and δi(K) ⊂ K for any 1 ≤ i ≤ n. We will need some facts about iterated polynomial rings. Remark 1.1. (1) When δ = 0 we denote the Ore extension as R = K[t; σ]. Similarly when σ = id, we write R = K[t; δ]. (2) We remark that if u ∈ K an invertible element and, for any x ∈ K, we have σ(x) = Iu(x) = uxu−1 then K[t; σ; δ] = K[u−1 t; u−1 δ]. Notice that if δ = 0, this gives a usual polynomial ring. (3) If σ is an endomorphism of the ring K and x, a are elements in K, we define δa(x) = ax − σ(x)a. This is a σ-derivation on K called the inner derivation on K. Let us notice that in this case we have, for any x ∈ K that (t − a)x = σ(x)(t − a) so that R = K[t; σ, δ] = K[t − a; σ]. (4) Combining the two previous remarks we have that, for any u, v ∈ K with u an invertible element, K[t; Iu, δv] = K[u−1 (t − v)] is a usual polynomial ring. Let us briefly recall the definition of the evaluation of a polynomial f(t) ∈ R = K[t; σ, δ] at an element a ∈ K. We define f(a) ∈ K to be the only element of K such that f(t) − f(a) ∈ R(t − a). This means, in particular, that a ∈ K is a right root of f(t) when t − a is right factor of f(t). We then introduce, for any i ≥ 0, a map Ni defined by induction as follows: N0(a) = 1, Ni+1(a) = σ(Ni(a))a + δ(Ni(a)). This leads to a concrete formula for the evaluation of any polynomial f(t) = n i=0 biti ∈ R = K[t; σ, δ] at an element a ∈ K: f(a) = n i=0 biNi(a). Before defining the evaluation of iterated polynomials, we now provide a few classical examples.
- 44. EVALUATION OF ITERATED ORE POLYNOMIALS 25 Example 1.2. (1) Consider the complex number C and the conjugation denoted by “−”. We then construct the Ore extension: R = C[t; −]; the commutation rule is here t(a + ib) = (a − ib)t, where a, b ∈ R. An element a ∈ C is a (right) root of t2 + 1 if N2(a) + 1 = 0 i.e. aa + 1 = 0. From this it is clear that t2 + 1 is an irreducible polynomial in R. On the other hand, it is easy to check that t2 + 1 is a central polynomial and we get that the quotient R/(t2 + 1) is isomorphic to the quaternion algebra H. On the other hand, the roots of the polynomial t2 − 1 are exactly the complex numbers of norm 1. (2) Another important kind of Ore extension is obtained by presenting the Weyl algebra A1 = k[X][Y ; id., d dx ], where k is a field. The commutation rule that comes up is thus Y X = XY + 1. In characteristic zero this algebra is simple (it doesn’t have any two sided ideal except 0 and A1). If char(k) = p 0, A1 is a p2 dimensional algebra over its center k[Xp , Y p ]. Evaluating powers of Y at X we get for instance Y 2 (X) = X2 + 1, Y 3 (X) = X3 + 3X, and Y 4 (X) = X4 + 6X2 + 3. We can iterate the procedure and get the Weyl algebra An(k). (3) In coding theory, Ore extensions of the form Fq[t; θ], where q = pn and θ is the Frobenius automorphism defined by θ(a) = ap , for a ∈ Fq, have been extensively used (cf. [2]). In [7], both the evaluation and the factor- ization for these extensions were described in term of classical (untwisted) evaluation of polynomials. General Ore extensions have also been used for constructing codes [3]. Let f(t1, t2, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] and an n-tuple (a1, a2, . . . , an) ∈ Kn be given (K is a ring which is assumed to be stable by the maps σi and δi, with1 ≤ i ≤ n). We want to define the value of f(a1, a2, · · · , an). It will be convenient to have notations for intermediate Ore extensions. So we put R0 = K, and, for i ∈ {1, . . . , n − 1}, Ri = Ri−1[ti; σi, δi]. Notice that Rn = R. Let (a1, a2, · · · , an) ∈ Kn and consider f0 = f(t1, . . . , tn) ∈ R = Rn−1[tn; σn, δn]. We can define f1 = f(t1, . . . , tn−1, an) ∈ Rn−1 as the remainder of the division of f0 on the right by tn − an. This is f0 = q(t1, . . . , tn−1)(tn − an) + f1. By the same procedure we divide f1 = f(t1, . . . , tn−1, an) by tn−1 − an−1 and get the remainder f2 = f(t1, . . . , tn−2, an−1, an) ∈ Rn−2. This is f1 = f(t1, . . . , tn−1, an) = q2(t1, t2, . . . , tn−1)(tn−1 − an−1) + f2. Continuing this process, we define f3 ∈ Rn−3, . . . fn−1 ∈ R1, fn ∈ R0 = K. The evaluation of f(t) at (a1, . . . , an) is fn ∈ K. This leads to the following definition. Definition 1.3. For f(t1, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] and (a1, . . . , an) ∈ Kn , we define f(a1, . . . , an), the evaluation of f(t1, t2, . . . , tn) at (a1, . . . , an), as the representative in K of f(t1, t2, . . . , tn) modulo In(a1, . . . , an) = R1(t1 −a1)+· · ·+Rn−1(tn−1 −an−1)+R(tn −an), where, for 1 ≤ i ≤ n, Ri stands for Ri = K[t1; σ1, δ1] · · · [ti; σi, δi]. Remark that In = In(a1, . . . , an) is an additive subgroup of R, + and is not, in general, an ideal in R = Rn.
- 45. 26 NABIL BENNENNI AND ANDRÉ LEROY Another way to compute the evaluation is to remark that the polynomials in R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] can be written in a unique way as sums of monomials of the form αl1,...,ln tl1 1 tl2 2 · · · tln n , for some 0 ≤ l1, l2, . . . , ln ≤ n and αl1,...,ln ∈ K. The sum n i=1 li is called the degree of the monomial and the degree of a polynomial is given by the degree of the monomials with higher degree. First remark that if a monomial m = m(t1, . . . , tn) is of degree l, then for any a ∈ K∗ , ma is a polynomial of degree l as well. We define the evaluation by induction on the degree. In practice, it is sufficient to define the evaluation of a monomial. For (a1, a2, . . . , an) ∈ Kn and m = m(t1, . . . , tn) we define m(a1, . . . , an) as follows: If deg(m) = 1 and m = αti for some 1 ≤ i ≤ n and α ∈ K then m(a1, . . . , an) = ai. So, assume that the evaluation of the monomials of degree l has been defined and consider a monomial m = m(t) such that deg(m) = l and m = m (t1, . . . , tj)tj for some 1 ≤ j ≤ n, then m(a1, . . . , an) = m (a1, . . . , aj) where the polynomial m (t1, . . . , tj) = m (t1, . . . , tj)aj is of degree smaller than l. Let us make some remarks about this evaluation. Remark 1.4. (1) First let us remark that, before we evaluate a polynomial, we must write it with the variables appearing in the precise order t1, t2, . . . , tn (from left to right). In other words, before evaluating a polynomial we must write it as a sum of monomials of the form tl1 1 tl2 2 · · · tln n . (2) Of course, this is not the only possible definition, but although it might look a bit strange, still it is natural if we want the zeros being right roots. Let us look more closely to the case of two variables. In other words it is natural for (a1, a2) to annihilate a polynomial of the form g(t1)(t2 − a2); but we don’t necessarily expect (a1, a2) to be a zero of a polynomial of the form (t1 − a1)h(t1). (3) Since the base ring is not assumed to be commutative we must be very cautious while evaluating a polynomial, even when the variables com- mute. With this definition, t1t2 ∈ K[t1, t2] evaluated at (a, b) gives (t1t2)(a, b) = ba. This might look very strange but if we think of evalua- tion in terms of “operators” via the right multiplication by b followed by the right multiplication by a this evaluation looks perfectly fine and the apparent awkwardness disappears. (4) We assumme that the different endomorphisms σis are such that σi(K) ⊆ K. In other words, for i 1, σi is an extension of σi|K to Ri−1. Some com- mutation relations exist between the different endomorphisms σi. For in- stance, consider the polynomial ring extension R = K[t1; σ1][t2; σ2]. If we put σ2(t1) = l i=0 aiti computing σ2(σ1(a)t1) = σ2(t1a) = σ2(t1)σ2(a), leads to the following equations ∀ 0 ≤ i ≤ l, aiσi 1(σ2(a)) = σ2(σ1(a))ai. Let us now give some examples. Example 1.5. (1) Let A1(k) = k[X][Y ; id., d dX ] and (a, b) ∈ K2 , then • Y X = XY + 1 = X(Y − b) + bX + 1 = X(Y − b) + b(X − a) + ba + 1 and hence (Y X)(a, b) = ba + 1.
- 46. EVALUATION OF ITERATED ORE POLYNOMIALS 27 • Y X2 = X2 Y + 2X = X2 (Y − b) + bX2 + 2X = X2 (Y − b) + bX(X − a) + baX + 2(X − a) + 2a and hence (Y X2 )(a, b) = ba2 + 2a. • Y 2 X = XY 2 + 2Y = XY (Y − b) + bX(Y − b) + bXb + 2(Y − b) + 2b and hence (Y 2 X)(a, b) = b2 a + 2b. (2) Consider the double Ore extension R = Fq[t1; θ][t2; θ], where q = pn , θ(a) = ap , and θ(t1) = t1. A polynomial p(t1, t2) ∈ R can be written as p(t1, t2) = n i=0 pi(t1)ti 2 = i,j ai,jtj 1ti 2 and we can easily check that p(t1, t2)(a, b) = i,j θj (Ni(b))Nj(a) = b (pi−1)pj p−1 a pi−1 p−1 . Let us now turn to another possible way of evaluating a polynomial f(t1, . . . , tn) ∈ R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] at (a1, a2, . . . , an) ∈ Kn . We consider the element of K representing f in the quotient R/I where I = R(t − a1) + R(t − a2) + · · · + R(t − an). The set I is the usual left ideal of R and this evaluation looks more classical. We now compare the two evaluations by comparing In = R1(t1 − a1) + · · · + Rn−1(tn−1 − an−1) + R(tn − an) and I. Theorem 1.6. Let K be a ring and R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn]. We consider (a1, a2, . . . , an) ∈ Kn and put I = R(t−a1)+R(t−a2)+· · ·+R(t−an) and In = R1(t1−a1)+· · ·+Rn−1(tn−1−an−1)+R(tn−an), where, for 1 ≤ i ≤ n, Ri, Ri = K[t1; σ1, δ1] · · · [ti; σi, δi]. With these notations, the following are equivalent: (1) In = I; (2) R(ti − ai) ⊆ In; (3) I = R; (4) For 1 ≤ i j ≤ n, we have tj(ti − ai) ∈ In; (5) For 1 ≤ i j ≤ n, we have σj(ti − ai)aj + δj(ti − ai) ∈ In; (6) For 1 ≤ i j ≤ n, we have (tjti)(a1, . . . , an) = σj(ai)aj + δj(ai). Proof. 1) ⇔ 2) is clear. 2) ⇒ 3) If I = R, then In = R and, for 1 ≤ i ≤ n, there exist polynomials gi(t1, t2 · · · , ti) ∈ Ri such that 1 = n i=1 gi(t1, . . . , ti)(ti − ai). The change of variables defined by putting yi = ti − ai gives that, for some hi ∈ Ri, we have 1 = n i=1 hi(y1, . . . , yi)yi. A comparison of the coefficients of degree zero of this equality leads to a contradiction. 3) ⇒ 4) There exists c ∈ K such that tj(ti − ai) − c ∈ In ⊆ I. Since tj(ti − ai) ∈ I we obtain c ∈ I hence, by 3), we must have c = 0. This shows that tj(ti − ai) ∈ In. 4) ⇒ 5) The fact that tj(ti − ai) ∈ In, implies that σj(ti − ai)tj + δj(ti − ai) ∈ In. Since (tj − aj) ∈ In, we obtain that σj(ti − ai)aj + δj(ti − ai) ∈ In. 5) ⇒ 6) This is clear since 5) implies that tjti − σj(ai)aj − δj(ai) = σj(ti)tj + δj(ti) − σj(ai)aj − δj(ai) ∈ In. 6) ⇒ 1) The equality in 6) implies that tjti−σj(ai)aj −δj(ai) ∈ In for 1 ≤ i j ≤ n. This gives that tj(ti −ai) ∈ In. Since also we have that Ri(ti −ai) ⊆ In we conclude that tj(ti − ai) ∈ In for any integer 1 ≤ i ≤ j ≤ n and hence R(ti − ai) ⊆ In for any 1 ≤ i ≤ n. This yields In = I, as required. Remark 1.7. (1) If n = 1, we obviously have I = I1 and all points are good. (2) In general, the two additive subset In ⊂ I are different. As mentioned above, we will use In for our evaluation. The reason is that while eval- uating with respect to I we often face the following problem: the left
- 47. 28 NABIL BENNENNI AND ANDRÉ LEROY R-module I can be the entire ring. So that the evaluation of any poly- nomial at (a1, a2 . . . , an) ∈ Kn with respect to I is zero. This the case in the Weyl algebra R = A1(K) = K[t1][t2; id, d dt1 ] for the piont (0, 0) since we then have t2t1 − t1t2 = 1 and hence Rt1 + Rt2 = R. This is not the case with our evaluation since, for instance, t2t1 = t1t2 + 1, so that t2t1 + I2(0, 0) = 1 + I2(0, 0) and hence the evaluation of t2t1 at (0, 0) is just 1. (3) In fact, it is quite often the case that I = R, even if we are using Ore polynomials with zero derivations. Consider for instance the Ore extension R = K[t1; σ1][t2; σ2] where K is a field and σ2 is an endomorphism of K[t1; σ1] such that σ2(t1) = t1. It is easy to check that for any (a1, a2) ∈ K2 we have (t2 − σ1(a2))(t1 − a1) + (−t1 + σ2(a1))(t2 − a2) = σ1(a2)a1 − σ2(a1)a2. So that if σ1(a2)a1−σ2(a1)a2 = 0, then the left ideal I(a1, a2) = R. This shows that very often the evaluation modulo I turns out to be trivial. Once again, this is not the case with our evaluation, since we have that t2(t1 − a1) is represented by σ1(a2)a1 − σ2(a1)a2 modulo I2(a1, a2). Definition 1.8. A point (a1, . . . , an) ∈ Kn will be called a good point if the two ways of evaluating a polynomial inK[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] at (a1, . . . , a2) coincide i.e. if In(a1, . . . , an) = I. The advantage of the good points is that in this case the evaluations via a left ideal I and In coincide. But of course, we can still evaluate a polynomial at any points via our additive subset In = R1(t1−a1)+· · ·+Rn−1(tn−1−an−1)+R(tn−an). Example 1.9. (1) In the classical case (σi = idK and δi = 0, for every 1 ≤ i ≤ n), every point (a1, . . . , an) ∈ Kn is good. (2) If K is a division ring σ1 = idK, δ1 = 0 and σ2 = id, δ2 = d/dt1, we have for any a, b ∈ K, (t2 − b)(t1 − a) = (t1 − a)(t2 − b) + 1. This shows that in this case there are no good points. Working with In instead of I we avoid the problem of having points that are zeroes of every polynomial in the ring R = K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn]. We will consider Reed-Muller codes and hence we will need to evaluate polyno- mials in several variables. We consider the case of iterated Ore polynomials defined on a finite base ring (field) K. In classical Reed-Muller coding the polynomials that are used to make evaluations are restricted to be of some bounded degree in each variables. Indeed polynomial maps associated to Xq − X ∈ Fq[X] are identically zero. So our first task is to consider what is the analogue of this polynomial in an Ore polynomial ring. We will need a few definitions. Definition 1.10. Let a ∈ K be an element of a division ring K, σ and δ an endomorphism of K and a σ-derivation of K, respectively. (1) For a nonzero x ∈ K, we denote ax = σ(x)ax−1 + δ(x)x−1 and put Δ(a) = {ax | x ∈ K∗ }. This set is called the (σ, δ) conjugacy class of a. (2) The map Ta : K −→ K defined by Ta(x) = σ(x)a + δ(x) is called the (σ, δ) pseudo-linear map associated to a. (3) The (σ, δ) centralizer of a is the set Cσ,δ (a) = {x ∈ K∗ | ax = a} ∪ {0}. Let us mention that the notion of σ, δ conjugation appears naturally due to the fact that while evaluating a product fg we have the nice formula fg(a) = 0 if g(a) = 0 and (fg)(a) = f(ag(a) )g(a) if g(a) = 0. Let us also mention that the
- 48. EVALUATION OF ITERATED ORE POLYNOMIALS 29 evaluation at an element a ∈ K is strongly related to the (σ, δ)-pseudo linear map Ta via the equality f(a) = f(Ta)(1) or more generally that, for any h ∈ R = K[t; σ, δ], a ∈ K and x ∈ K∗ , h(Ta)(x) = h(ax )x. We first need the following lemma which is part of folklore. Theorem 1.11. Let a be an element of a division ring K and f(t) ∈ R = K[t; σ, δ] be a polynomial of degree n. Then: (1) Cσ,δ (a) is a subdivision ring of K. (2) The map Ta is a right Cσ,δ (a) linear map. (3) f(t) has roots in at most n (σ, δ)-conjugacy classes, say {Δ(a1), . . . , Δ(ar)}, r ≤ n; (4) r i=1 dimC(ai) ker(f(Tai )) ≤ n, where C(ai) := Cσ,δ (ai) for 1 ≤ i ≤ r. Proof. We refer the reader to [8] and [11]. Notice that point (4) above generalizes several classical theorems, in particular, the theorem by Gordon-Motzkin that states that a polynomial f(X) ∈ K[X] with coefficients in a division ring K can have roots in at most n = deg(f) classical (σ = id., δ = 0)-conjugacy classes. Theorem 1.12 below can be found in [7]. Theorem 1.12. Let p be a prime number and Fq be the finite field with q = pn elements. Denote by θ the Frobenius automorphism. Then: (a) There are p distinct θ-conjugacy classes in Fq. (b) Cθ (0) = Fq and, for 0 = a ∈ Fq, we have Cθ (a) = Fp. (c) In Fq[t; θ], the least left common multiple of all the elements of the form t − a for a ∈ Fq is the polynomial G(t) := t(p−1)n+1 − t. In other words, G(t) ∈ Fq[t; θ] is of minimal degree such that G(a) = 0 for all a ∈ Fq. (d) The polynomial G(t) obtained in c) above is invariant, i.e. RG(t) = G(t)R. Proof. (a) Let us denote by g a generator of the cyclic group F∗ q := Fq {0}. The θ-conjugacy class determined by the zero element is reduced to {0} i.e. Δ(0) = {0}. The θ-conjugacy class determined by 1 is a subgroup of F∗ q: Δ(1) = {θ(x)x−1 | 0 = x ∈ Fq} = {xp−1 | 0 = x ∈ Fq}. It is easy to check that Δ(1) is cyclic generated by gp−1 and has order pn −1 p−1 . Its index is (F∗ q : Δ(1)) = p − 1. Since two nonzero elements a, b are θ-conjugate if and only if ab−1 ∈ Δ(1), we indeed get that the number of different nonzero θ-conjugacy classes is p − 1. This yields the result. (b) If a ∈ Fq is nonzero, then Cθ (a) = {x ∈ Fq | θ(x)a = ax} i.e. Cθ (a) = Fp. (c) We have, for any x ∈ Fq, (t(p−1)n+1 − t)(x) = θ(p−1)n (x) . . . θ(x)x − x. Since θn = id, and Nn(x) := θn−1 (x) . . . θ(x)x ∈ Fp, we get (t(p−1)n+1 − t)(x) = x(θn−1 (x) . . . θ(x)x)p−1 − x = xNn(x)p−1 − x = 0. This shows that indeed G(t) annihilates all the elements of Fq and hence G(t) is a left common multiple of the linear polynomials {(t − a) | a ∈ Fq}. Let h(t) := [t − a | a ∈ Fq]l denote their least left common multiple. It remains to show that deg h(t) ≥ n(p − 1) + 1. Let 0 = a0, a1, . . . , ap−1 ∈ Fq be elements representing the θ-conjugacy classes (Cf. a) above). Denote by C0, C1, . . . , Cp−1 their respective θ-centralizer. The formulas recalled in the paragraph before Theorem 1.11 shows that h(Ta)(x) = h(ax )x = 0 for any nonzero element x ∈ Fq and any element a ∈ {a0, . . . , ap−1}. Hence ker h(Tai ) = Fq for 0 ≤ i ≤ p − 1. Using Inequality (4) in Theorem 1.11 and
- 49. 30 NABIL BENNENNI AND ANDRÉ LEROY the statement b) above, we get deg h(t) ≥ p−1 i=0 dimCi ker h(Tai ) = dimFq Fq + p−1 i=1 dimFp Fq = 1 + (p − 1)n, as required. (d) Since θn = id., we have immediately that G(t)x = θ(x)G(t) and obviously G(t)t = tG(t). Let us now consider an iterated Ore extension K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn], where K is a finite ring. Using classical notations from the world of algebraic ge- ometry, we consider I(Kn ) = {f(t1, . . . , tn) ∈ R | ∀(a1, . . . , an) ∈ Kn , f(a1, . . . , an) = 0}. In other words: I(Kn ) = ∩(a1,...,an)In(a1, . . . , an). In the case of an Ore extension Fq[t1; θ] . . . [tn; θ], where θ is the Frobenius automorphism of Fq = Fpn , the above theorem shows that, for any 1 ≤ i ≤ n, t (n−1)p i − ti ∈ I(Kn ). Having this in mind we introduce, for any 1 ≤ i ≤ n, the monic polynomials Gi = Gi(t1, . . . ti) of minimal degree in Ri = K[t1; σ1, δ1] . . . [ti; σi, δi] such that Gi ∈ I(Kn ). With these notations, we can now state the following proposition. Proposition 1.13. Let K be a finite ring and, for 1 ≤ i ≤ n, consider the polynomials Gi ∈ Ri = K[t1, σ1, δ1] . . . [ti, σi, δi] defined above. Then I(Kn ) = R1G1(t1) + R2G2(t1, t2) + · · · + RnGn(t1, . . . , tn). Proof. It is clear that R1G1(t1) + R2G2(t1, t2) + · · · + RnGn(tn) ⊆ I(Kn ). Now, if a polynomial f ∈ K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] belongs to I(Kn ), then, for any (a1, . . . , an) ∈ Kn , f(a1, a2, . . . , an) = 0. Let us write the consecutive remainders obtained during the evaluation process as follows. For any (a1, . . . , an) ∈ Kn , we have f(t1, . . . , tn) = q1(t1, . . . , tn)(tn − an) + f1(t1, . . . , tn−1, an). f1(t1, . . . , tn−1, an) = q2(t1, . . . , tn−1)(tn−1 − an−1) + f2(t1, . . . , tn−2, an−1, an). Continuing this process, we get fn−2(t1, t2, a3, . . . , an) = qn−1(t1, t2)(t2 − a2) + fn−1(t1, a2 . . . , an) and fn−1(t1, a2 . . . , an) = qn(t1)(t1 − a1) + fn(a1, . . . , an−1, an). Since f(a1, a2, . . . , an) = fn(a1, . . . , an−1, an) = 0, we conclude that for any for any (a1, . . . , an ∈ Kn , fn−1(t1, a2 . . . , an) ∈ R1(t1−a1), i.e. that fn−1(t1, a2 . . . , an) ∈ R1G1(t1). Similarly, we have that fn−2(t1, t2, a3, . . . , an) − fn−1(t1, a2 . . . , an) ∈ R2 ∩ I(Kn ) = R2G2(t1, t2). An easy induction yields that for any 1 ≤ i ≤ n, we have fn−i(t1, . . . ti, ai+1, . . . , an) − fn−i+1(t1, . . . , ti−1, ai, . . . , an) ∈ RiGi. From this we get successively that f ∈ RnGn + f1 ⊆ RnGn + Rn−1Gn−1 + f2 ⊆ RnGn + Rn−1Gn−1 + Rn−3Gn−3 + f3 ⊆ R1G1(t1) + R2G2(t1, t2) + · · · + RnGn(t1, . . . , tn). This yields the desired result. 2. Skew Reed-Muller codes using iterated skew polynomial rings In the first part of this section, we give some preliminaries on Reed-Muller codes over the commutative polynomial ring. Let m be a positive integer and let P1, P2, . . . , Pn be the n = qm points in the affine space Am (Fq) . For any integer r with 0 ≤ r ≤ m(q − 1), let Rm ⊆ Fq [x1, x2, . . . , xm] /(xp 1 − x1, xp 2 − x2, · · · , xp m − xm) be the set representing polyno- mials of degree less or equal than r.
- 50. EVALUATION OF ITERATED ORE POLYNOMIALS 31 Definition 2.1. Let f = n i=0 aIXI be a polynomial in Fq [x1, x2, . . . , xm], where for I = (i1, . . . , in) we define XI to be xi1 xi2 · · · xin . The operator evaluation ev(f) at b = (b1, b2 . . . bn) ∈ Fn q is defined as Rm → Fq ev : f = i∈I aiXi → f(b) = i∈I ai(bi ). The Reed-Muller codes of the length n = qm can be obtained as evaluation of the polynomial f in Rm in all points P1, P2, · · · , Pn in the affine sapce Am (Fq) . The Reed-Muller codes of order r is defined as Rq(r, m) = {(f (P1) , f (P2) , . . . , f (Pn)) | f ∈ Rm} = {ev(f) : f ∈ Rm, deg f ⩽ r} . The function ev : Rm → Fq is an homomorphism of algebras (see [1]). Hence the Reed-Muller codes is generated by the codewords ev(f) where f is a monomial of degree less than or equal to r in the set of monomials in m variables. It will be denoted M = {Xa1 1 Xa2 2 . . . Xan n } . 2.1. Iterated skew polynomial and Reed-Muller codes. In a recent pa- per, skew Reed-Muller codes were defined via the use of iterated Ore extensions with three variables over a finite field [5]. The authors used inner derivations and inner automorphisms, and (cf. Remarks 1.1) these can be erased by a change of variables. Their idea was to used Gröbner bases to make computations. They used the available software on the subject. But their methods used the ideal I in our notations and hence they faced the problem that I might be equal to the whole iterated Ore extension R. Some of their computations were wrong and for the sake of future works, we will briefly correct them. In the lines below the variables Xi are the one used in their paper. 1. Let F4 = F2(α), where α2 = α + 1. In the ring R1 = F4 [Y1; θ] where θ is the Frobenius automorphism, we put X1 = Y1 + 1 and we have the commutation relation X1α = (Y1 + 1)α = θ(α)Y1 + α = θ(α)X1 + θ(α) + α = α2 X1 + 1. 2. In the ring R2 = R1 [Y2] we put X2 = α2 Y2 + Y1 + α2 so that we have R2 = R1[X2; θα, δX1+α], where θα(x) = α−1 xα for any x ∈ R1. In R2 we have the above commutation relation X1α = α2 X1 + 1 together with X2α = αX2 + X1 + 1 and X2X1 = αX1X2 + α2 X2 + α2 X2 1 + α2 . 3. In the ring R3 = R2[Y3], we put X3 = α2 Y3 + αY1 + α. We then have R3 = R2 [X3; θα, δαX1 ] where, as above θα(x) = α−1 xα, for any x ∈ F4. We have the above commutation relations together with X3α = αX3 + αX1 + α, X3X1 = αX1X3 + α2 X3 + α2 X2 1 + α2 X1, X3X2 = X2X3 + α2 X1X3 + α2 X3 + X1X2 + αX2 1 + α2 X1 + X2 + 1. Of course, the monomials in X1, X2, X3 are not monomials in Y1, Y2, Y3 and con- versely. The advantage of the variables Y1, Y2, Y3 is clear. We continue with these variables and consider,
- 51. 32 NABIL BENNENNI AND ANDRÉ LEROY Rn = Rn−1[Yn] = R1[Y2, Y3, . . . , Yn] = F4[Y1, θ][Y2, Y3, . . . , Yn]. As in the commutative case, there are polynomials that annihilate any element of Fn q . Thanks to Theorem 1.12 and 1.13, we have that, in our situation I(Kn ) = R1G1 + R2G2 + · · · + RnGn where G1 = Y 3 1 − Y1 and Gi = Y 4 i − Yi for any 2 ≤ i ≤ n (notice that the form of G1 is due to the theorem 1.12). This iterated skew polynomial ring leads to a new family of skew Reed-Muller code generated by the evaluations of a base of monomials in m variables and of degree less than or equal to r given by M = Y l1 1 Y l2 2 . . . Y ln n . By iterated skew polynomial rings given above we have the evaluation of f ∈ Rm at b = (b1, b2 . . . bn) ∈ Fn q is defined as Rm → Fq ev : f(X1, X2, . . . , Xn) → f(b1, . . . , bn) = n i=0 P(b2, . . . , bn)θ i(i+1) 2 (b1). Where f(X1, X2, . . . , Xn) = g(Y1)P(Y2, . . . , Yn) ∈ R1G1 + R2G2 + · · · + RnGn. By the iterated skew polynomial ring and the evaluation we have the new new family of the skew Reed-Muller codes in the following proposition. Proposition 2.2. The skew Reed-Muller codes of parameters r and m over Rm = Fq[X1; θ1, δ1] . . . [Xm; θm, δm] are generated as R(r, m) def = n i=0 P(b2, . . . , bn)θ i(i+1) 2 (b1) : P ∈ Rm such that deg P ⩽ deg Gn . The choice of the rows used to form the matrix generating the Reed-Muller codes R(r, m) consists in selecting the monomials Y l1 1 Y l2 2 . . . Y ln n of degree less than or equal to r = (n−1)p+1. This choice is based on the order of the monomials. The group of permutations of Reed-Muller codes R(r, m) is the set of transformations affines x → Ax + b where A ∈ Fm×m q is an invertible matrix and b ∈ Fm q . 2.2. Skew Reed-Muller codes using polynomial maps in n variables. In this section we give the construction of skew Reed-Muller codes using polynomial maps in n variables. We know that the Reed-Muller codes are defined as the codes generated by the evaluations of a base of monomials in m variables and of degree less than or equal to r, the set of monomials in m variables will be denoted M = tl1 1 tl2 2 . . . tln n . Notice that in the case of iterated Ore extensions K[t1; σ1, δ1][t2; σ2, δ2] · · · [tn; σn, δn] it is always possible to order the indeterminates of a word in t1, . . . , tn so that the only words we have to consider are the one of the form tl1 1 tl2 2 . . . tln n . We now give the evaluation of some monomials in R = K[t1, σ1, δ1][x2, σ2, δ2], · · · , [xn, σn, δn] at (a1, a2, · · · , an) ∈ Kn . (1) Consider the polynomial ring R = K[t1; σ1, δ1][t2; σ2, δ2] and let us evalu- ate the polynomials t1t2 at (a1, a2) ∈ K2 . We have t1t2 = t1(t2 − a2) + t1a2 = t1(t2 − a2) + σ1(a2)t1 + δ1(a2). This leads to (t1t2)(a1, a2) = σ1(a2)a1 + δ1(a2).
- 52. EVALUATION OF ITERATED ORE POLYNOMIALS 33 (2) Consider the polynomial ring R = K[t1; σ1, δ1][t2; σ2, δ2][t3; σ3, δ3] and let us evaluate the polynomials t1t2t3 at (a1, a2, a3) ∈ K3 . We have t1t2t3 = t1t2(t3 − a3) + t1t2a3 = t1t2(t3 − a3) + t1(σ2(a3)t2 + δ2(a3)) = t1t2(t3 − a3)+t1σ2(a3)t2 +t1δ2(a3) = t1t2(t3 −a3)+σ1(σ2(a3))t1t2 +σ1(δ2(a3))t1 + δ1(δ2(a3)). This leads to (t1t2t3)(a1, a2, a3) = σ1(σ2(a3))a1a2 + σ1(δ2(a3))a1 + δ1(δ2(a3)). (3) Consider the polynomial ring R = K[t1; σ1, δ1][t2; σ2, δ2][t3; σ3, δ3][t4; σ4, δ4]. We have (t1t2t3t4)(a1, a2, a3, a4) = σ1(σ2(σ3(a4)a3)a2)a1 + δ1(σ2(σ3(a4)a2)a3) + σ1(δ2(σ3(a4)a3))a1 +δ1(δ2(σ3(a4)a3))+σ1(σ2(δ3(a4))a2)a1 +δ1(σ2(δ3(a4))a2)+ σ1(δ2(δ3(a4)))a1 + δ1(δ2(δ3(a4)))). Polynomial maps in n variables and the evaluation leads to the new family of skew Reed-Muller codes in the following example. Example 2.3. In this example, we give the construction of the Reed-Muller codes with polynomial maps in 2 or 3 variables. In these examples we used the Frobenius automorphsim for σ1 and the identity for σ2 and σ3. The derivations are all zeroes. (1) The evaluation of the set monomial {1, t1, t2, t1t2} over R = F4[t1; σ1, δ1][t2; σ2, δ2] gives the Reed-Muller code with parameters [16, 4, 8]. (2) The evaluation of the set monomial {1, t1, t2, t2t1} over R = F4[t1; σ1, δ1][t2; σ2, δ2] gives the Reed-Muller code with parmeters [16, 4, 7]. (3) The evaluation of the set monomial {1, t1, t2, t1, t3, t1t3, t2t3, t1t2t3} over R = F4[t1; σ1, δ1][t2; σ2, δ2][t3; σ3, δ3] gives the Reed-Muller code with pa- rameters [64, 8]. References [1] M. Bardet, V. Dragoi, A. Otmani, and J.P. Tillich, Algebraic properties of polar codes from a new polynomial formalism, IEEE International Symposium on Information Theory (ISIT), 230–234, 2016. [2] Delphine Boucher and Felix Ulmer, Coding with skew polynomial rings, J. Symbolic Comput. 44 (2009), no. 12, 1644–1656, DOI 10.1016/j.jsc.2007.11.008. MR2553570 [3] M’Hammed Boulagouaz and André Leroy, (σ, δ)-codes, Adv. Math. Commun. 7 (2013), no. 4, 463–474, DOI 10.3934/amc.2013.7.463. MR3119685 [4] I. Dumer, R. Fourquet, and C. Tavernier, Cryptanalysis of Block Ciphers via Decoding of Long Reed-Muller Codes, Algebraic and Combinatorial Coding Theory (ACCT 2010), Novosibirsk, Russia September 2010. [5] Willi Geiselmann and Felix Ulmer, Skew Reed-Muller codes, Rings, modules and codes, Con- temp. Math., vol. 727, Amer. Math. Soc., [Providence], RI, [2019] c 2019, pp. 107–116, DOI 10.1090/conm/727/14628. MR3938143 [6] Tadao Kasami, Shu Lin, and W. Wesley Peterson, New generalizations of the Reed- Muller codes. I. Primitive codes, IEEE Trans. Inform. Theory IT-14 (1968), 189–199, DOI 10.1109/tit.1968.1054127. MR275989 [7] André Leroy, Noncommutative polynomial maps, J. Algebra Appl. 11 (2012), no. 4, 1250076, 16, DOI 10.1142/S0219498812500764. MR2959425
- 53. 34 NABIL BENNENNI AND ANDRÉ LEROY [8] T. Y. Lam, A. Leroy, and A. Ozturk, Wedderburn polynomials over division rings. II, Non- commutative rings, group rings, diagram algebras and their applications, Contemp. Math., vol. 456, Amer. Math. Soc., Providence, RI, 2008, pp. 73–98, DOI 10.1090/conm/456/08885. MR2416145 [9] T. Y. Lam and André Leroy, Principal one-sided ideals in Ore polynomial rings, Algebra and its applications (Athens, OH, 1999), Contemp. Math., vol. 259, Amer. Math. Soc., Providence, RI, 2000, pp. 333–352, DOI 10.1090/conm/259/04106. MR1780532 [10] D.E. Muller, Application of Boolean algebra to switching circuit design and to error detection, Transactions of the IRE professional group on electronic computers, vol.3 (1954), 6–12. [11] André Leroy and Adem Ozturk, Algebraic and F-independent sets in 2-firs, Comm. Algebra 32 (2004), no. 5, 1763–1792, DOI 10.1081/AGB-120029901. MR2099700 USTHB, Faculté de Mathématiques, BP 32 El Alia Bab Ezzouar, Algiers, Algeria Email address: nabil.bennenni@gmail.com Université Artois, UR 2462, Laboratoire de Mathématiques de Lens (LML), F-62300 Lens, France Email address: andre.leroy@univ-artois.fr
- 54. Contemporary Mathematics Volume 785, 2023 https://doi.org/10.1090/conm/785/15773 Identities of the algebra O ⊗ O Guy Blachar, Louis H. Rowen, and Uzi Vishne To Professor Syed Tariq Rizvi, for his contribution to Ring and Module Theory Abstract. We find nonassociative polynomial identities of minimal degree, which is 7, for the algebras O⊗O and M2(O), where O is the octonion algebra. 1. Introduction One of the basic invariants of an algebra A satisfying a polynomial identity (PI) is the minimal degree of an identity, which we call the PI degree. A commutative algebra has PI degree 2, because it satisfies [x1, x2]. That the PI degree of the matrix algebra Mn(F) is 2n is the Amitsur-Levitzki theorem. The Grassmann algebra satisfies [x1, [x2, x3]], so its PI degree is 3. The same notion applies to nonassociative algebras. The substitution in an associative polynomial is not well defined if the algebra is not associative, so an identity of A would be an element in the free nonassociative algebra, which vanishes under every substitution from A. Again we are concerned with the minimal degree of an identity. The associator (x1, x2, x3) = (x1x2)x3 − x1(x2x3) is an identity in the nonassociative variety precisely when the algebra is associative. The notion of polynomial identities, and thus the PI degree, depends on the variety. If V is a variety of algebras and A ∈ V, the nontrivial polynomial identities of A with respect to V are nonzero elements of the free algebra in V, so one excludes consequences of the defining identities of V. Let O denote the split octonion algebra over a base field F. The alternative PI degree of the octonion algebra is 5 (see Section 3 below), because the identities of lower degree, namely the consequences of (x, x, y) and (y, x, x), hold in any alternative algebra. The purpose of this note is to attract attention to the following question: Question 1.1. What is the (nonassociative) PI degree of the matrix algebras Mn(O)? What is the PI degree of the tensor power O⊗n ? Here we are concerned with the (general) nonassociative identities, as the cen- tral simple algebras Mn(O) and O⊗n are not alternative (for n 1); indeed, the only simple alternative algebras are associative or the octonion algebras. 2020 Mathematics Subject Classification. Primary 17A30; Secondary 17D05, 16R99. Partially supported by ISF grant #1994/20. c 2023 American Mathematical Society 35
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