Efficient Algorithm for Constructing KU-algebras from Block Codes

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 5 Issue 5|| May 2016 || PP.32-43
www.ijesi.org 32 | Page
Efficient Algorithm for Constructing KU-algebras from Block
Codes
Samy m.mostafa1
, Bayumy A.B.Youssef 2
, Hussein Ali Jad 2
1
(Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt)
2
(Informatics Research Institute, City for Scientific Research and Technological Applications, Borg El Arab,
Alexandria, Egypt)
ABSTRACT. In this paper, we will provide an algorithm which allows us to find a KU-algebra starting from a
given binary block code.
KEYWORDS: KU-algebras, Block codes, partially ordered set.
I. INTRODUCTION
The notion of BCK and BCI-algebras are first introduced by Imai and Iséki [6]. Later on, in 1984, Komori [8]
introduced a notion of BCC-algebras, and Dudek ([3], [13]) redefined the notion of BCC-algebras by using a
dual form of the ordinary definition in the sense of Komori. Accordingly, Dudek and Zhang [9] introduced a
new notion of ideals in BCC-algebras and described connections between such ideals and congruences.
Prabpayak and Leerawat [14] introduced a new algebraic structure which is called KU-algebra. They gave the
concept of homomorphisms of KU-algebras and investigated some related properties. For more details, see
([13], [14]). Over the last 70 years, algebraic coding has become one of the most important and widely applied
aspects of abstract algebra. Coding theory forms the basis of all modern communication systems, and is the key
to another area of study, Information Theory, which lies in the intersection of probability and coding theory.
Algebraic codes are now used in essentially all hardware-level implementations of smart and intelligent
machines, such as scanners, optical devices, and telecom equipment. It is only with algebraic codes that we are
able to communicate over long distances, or are able to achieve megabit bandwidth over a wireless channel.
Coding theory is the study of methods for efficient and accurate transfer of information from one party to
another. Various type of codes and their connections with other mathematical objects have been intensively
studied. The idea of coding theory is to give a method of how to convert the information into bits, such that
there are no mistakes in the received information, or such that at least some of them are corrected. On this
account, encoding and decoding algorithms are used to convert and reconvert these bits properly. In Coding
Theory, a block code is an error-correcting code which encodes data in blocks. In the paper [7], the authors
introduced the notion of BCK-valued functions and investigate several properties. Moreover, they established
block-codes by using the notion of BCK-valued functions. They show that every finite BCK-algebra determines
a block-code constructed a finite binary block-codes associated to a finite BCK-algebra. In [5] provided an
algorithm which allows to find a BCK-algebra starting from a given binary block code. In [16] the authors
presented some new connections between BCK- algebras and binary block codes. Mostafa et al in [10] applied
the code theory to KU- algebras and obtained some interesting results. In this paper, we provided an algorithm
which allows to find a KU-algebra starting from a given binary block code.
II. Preliminaries
Now, we recall some known concepts related to KU-algebra from the literature, which are helpful in further
study of this article.
Definition 2.1([13], [14]). (KU-algebra) Let X be a nonempty set with a binary operation and a constant 0 .
The triple )0,,( X is called a KU-algebra, if for all Xzyx ,, the following axioms are satisfied.
( 1
ku ) 0)]())[()(  zxzyyx ,
( 2
ku ) 00 x ,
( 3
ku ) xx 0 ,
( 4
ku ) 0 yx and 0 xy implies yx  ,
( 5
ku ) 0 xx .

Efficient Algorithm for Constructing KU-algebras…
On a KU-algebra X we can define a binary relation  on X by putting 0 xyyx . Then
),( X is a partially ordered set and 0 is its smallest element. Thus )0,,( X satisfies the following conditions.
For all Xzyx ,,
(
1
ku ) )()()( yxzxzy  ,
(
2
ku ) x0 ,
(
3
ku ) xyyx  , implies yx  ,
(
4
ku ) xxy  .
Theorem2.2 ([13], [14]). In a KU-algebra X . The following axioms are satisfied. For all Xzyx ,, ,
(1) yx  imply zxzy  ,
(2) )()( zxyzyx  , for all Xzyx ,, ,
(3) yxxy  ))(( .
Example 2.3. Let X = {0, 1, 2, 3, 4} be a set with a binary operation  defined by the following table
* 0 1 2 3 4
0 0 1 2 3 4
1 0 0 0 3 0
2 0 1 0 3 0
3 0 0 0 0 0
4 0 1 0 3 0
Table (1)
Then )0,,( X is a KU-algebra.
Definition 2.4[6]. (Sub-algebra) A non-empty subset S of a KU-algebra )0,,( X is called KU-sub algebra of
X if Syx  whenever Syx , .
Definition 2.5[5]. A KU-algebra(X, ∗, 0) is said to be KU - commutative if it satisfies: ∀x, y∈ X, (y∗ x) ∗ x =
(x∗ y) ∗ y.
Definition 2.6[11]. A KU-algebra(X, ∗, 0) is said to be KU -positive implicative, if it satisfies: (z∗ x) ∗ (z∗ y) =
z∗ (x∗ y), for all x, y, z in X.
Definition 2.7[11]. A KU-algebra(X, ∗, 0) 0is called KU- implicative if x = (x∗ y) ∗ x, for all x, y in X.
Definition 2.8[14]. (Homomorphism) Let )0,,( X and )0,,(  X be KU-algebras, a homomorphism is a
map XXf : satisfying )()()( yfxfyxf  for all Xyx , .
Theorem 2.9[14]. Let f be a homomorphism of KU-algebra X into KU-algebra X  . It follows that
(i) If 0 is the identity in X then )0(f is the identity in X  .
(ii) If S is a KU-subalgebra of X then )( Sf is a KU-subalgebra of X  .
(iii) If S is a KU-subalgebra of X  then )(
1
Sf

is a KU-subalgebra of X .

Definition 2.10. (Lexicographic) Let the two posets ),(),( 2211
 SandS . The lexicographic order  on
the Cartesian product 21
SS  is defined by specifying that one pair is less than the other pair,
i.e. ),(),( 2121
yyxx  iff 22211111
yxandyxoryx 
We obtain a partial ordering  by adding equality to the ordering  on 21
SS  .
Example 2.11. (Lexicographic Order) Let  ......,,,1
cbaS  and 1
 be the usual alphabetic order. Let
 10....,,3,2,1,02
S and 2
 be the usual alphabetic order be the usual partial order  ,
then 104,,)10,()4,(sin,)2,()8,( 2211
 aaaaandbaceba .
Now, we use some results literature from paper [10].
In what follows let A and X denote a nonempty set and a KU-algebra respectively, unless otherwise specified.
Definition 2.12. A mapping XAA :
~
is called a KU-valued function (briefly, KU-function) on A .
Definition 2.13. A cut function of A
~
, for Xq  is defined to be a mapping }1,0{:
~
AAq
such
that 0*)(
~
1)(
~
)(  qxAxAAx q
.
Obviously, q
A
~
is the characteristic function of the following subset of A , called a cut subset or a q-cut of A
~
.
Example 2.14. Let A ={x, y, z} and let X = {0, a, b, c, d} is a KU-algebra with the following Cayley table:
* 0 a b c d
0 0 a b c d
a 0 0 b b a
b 0 a 0 a d
c 0 0 0 0 a
d 0 0 b b 0
Table (2)
The function XAA :
~
given by 








cba
zyx
A
~
is a KU-function on A , and its cut subsets are
     xAAAyAxAA dcba
 ,,,,0
Let  yxAyx 

; ; for any Ax  ,

x is called equivalence class containing x .
Lemma 2.15. Let XAA :
~
be a KU- function on A . For every Ax  , we have  
 xxA inf)(
~
, that is
)(
~
xA the least element of the  to which it belongs.
Definition 2.16. Let  nA ,.....,3,2,1 and X be a finite KU-algebra. Then every KU-function
XAA :
~
on A determines a binary block code V of length n in the following way: To every

x ,
where Ax  , there corresponds a codeword nx
xxxV .....21
 Such that
 1,0)(
~
 jandAiforjiAxx xji
.
Let nx
xxxV .....21
 , ny
yyyV .....21
 be two code words belonging to a binary block-codeV .
Define an order relation c
 on the set of code words belonging to a binary block- code V as follows:
niforyxVV iiycx
,....,2,1 …………….………………….…… (1)

III. Basic Results
Suppose that (X,  ) be a finite partial ordered set with the minimum element . We define a binary relation
 on X as follows:








.,)3(
,,,)2(
,,,)1(
otherwiseyyx
Xyxxyifyx
Xxxxxx


…………………… (2)
Proposition3.1. The algebra (X,*, ) is a KU-algebra through the previous notations.
Proof: Conditions )( 2
ku , )( 3
ku , )( 4
ku and )( 5
ku are satisfied. Now, we prove condition )( 1
ku that
is 0=z))*(x*z)*((y*y)*(x , for all x, y, z ∈ X.
Case (1): at least one element is θ.
(1) ;x   *)*(*=z))*(*z)*((y*y)*( yzzy ,
(2) ;y   *)*(*=z))*(x*z)*((*)*(x zz ,
(3) .*)*()*(*)*(=))*(x*)*((y*y)*(x;z   yxyx
Case (2): one element is comparable with another.
(1) x ≤ y; ,*)*(*=z))*(x*z)*((y*y)*(x   yzzy
(2) x ≤ z; ,*)*(*=z))*(x*z)*((y*y)*(x   yzzy
(3) y ≤ x; ,*)*(*=z))*(x*z)*((y*y)*(x  zz
(4) y ≤ z; .*)*(*=z))*(x*z)*((y*y)*(x   yzzy
(5) z ≤ x; ,*)*(*=z))*(x*z)*((y*y)*(x   yzy
(6) z ≤ y; ,*)*(*=z))*(x*z)*((y*y)*(x   zyzy
Case (3): two elements are comparable with the third.
x ≤ y and z ≤ y;   zyzy *)*(*=z))*(x*z)*((y*y)*(x , etc.
Proposition3.2. a KU-algebra (X,*, 0) through the previous notations, is a non-positive implicative algebra.
Proof: we must prove condition in the above definition (2.6), that
is Xzyxallforyxzyzxz  ,,,)*(*)*(*)*( .
Case (1): at least one element is θ.
(1) ;x ,*)*(**)*(*)*( yyzyzandyyyzz  
(2) ;y ,*)*(**)*(*)*(   zxzandxzxz
(3) ;z ,*)*(**)*(*)*( yyzyzandyyyzz  
Case (2): one element is comparable with another.
(1) x ≤ y; ,*)*( yyzandyyx 
(2) x ≤ z; ,*)*( yyzandyy 
(3) y ≤ x; .*)*(   zandyyx
We have in Case 2 R.H.SL.H.S  ,then a KU-algebra (X,*, 0) is a non-positive implicative algebra.
Proposition3.3. a KU-algebra (X,*, 0) through the previous notations, is an implicative and non-commutative.
We denote a KU-algebra with n
C , if it has n elements. Suppose that V is a binary block code with n
codewords of length n, then we have v
M (the related matrix of the code V), where
})1,0({)( },....,2,1{,, nnjijiv
MmM  
with rows containing the codewords of V.
Theorem3.4. Let a matrix V
M be a lower triangular with },....,2,1{,1 nim ii
 , and
}1,0{......;......1 inikinik
xxxx in V. through the previous notations, we have a set A with n
elements, a KU-algebra X, and a KU-function XA:f  such that f determines V.

Proof. We define on V the lexicographic order that denoted by lex
 ,so we have (V, lex
 ) is a
totally ordered set. Suppose that V = { m
www ,......,, 21
}, with nlexlexlex
www  .........21
,
then 00.....101
w (number of zeros are (n-1) times) and 1.....1 imikn
xxw  ; imik
xx ..... ∈{0,1}.
We define also a partial order  on , then (V ,≤) is a partial ordered set with i
ww 1
,
i∈{1,2,……,n}, then 01
w and n
w is the maximal element in(V). If we define a binary relation
]2[ on (V, ≤) as in Proposition 3.3. We have ),,( 1
wVX  as a KU-algebra and V is isomorphic
to n
C as KU-algebras. Then we consider VA  and f: A→V , f (w) = w be the identity map as a
KU-function, then f provides a family of binary block code
,1)(};1,0{;{  xfAfV rrc n
if and only if XrAxrxf  ,,0)( .
Suppose that },1,0{.......,0000.......1,1,  iLikiLikkk
xxwherexxwnkVw  and number of
zeroes are (k-2).
If 





comparedbetcanwwwwww
wwww
x
jikjikjik
jikkji
ji
',0
01
A binary block code as in the previous Theorem can be generated by two or more algebras (see
examples 3.5, 3.6, and 3.7). But a KU-algebra generates a unique binary block code using the
algorithm in [10].
Example 3.5. Let V = {1001, 1100, 1110, 1000} be a binary block code, using the lexicographic order, the code
V can be written V = {1000, 1100, 1110, 1001} = },,,{ 4321
wwww . With the following graph
Fig. (1): A graph of code with 5 vertices and 7 edges.
In figure (1): },,,{ 4321
wwww are the set of vertices and { },{},,{},,{ 413221
wwwwww } are the set
of edges.
By using the previous theorem we define the partial order ≤ on V, then we get ,1 i
ww  i∈{2,3,4} ,
32
ww  , 2
w can’t be compared with 4
w , and 3
w can’t be compared with 4
w .then the operation [*] on V is
defined by the following table:
Table (3)
Then, V with the operation [*] is a KU-algebra. The same binary block code V can be obtained from a KU-
algebra ),,( A
1
w 2
w 3
w 4
w
1
w 1
w 2
w 3
w 4
w
2
w 1
w 1
w 3
w 4
w
3
w 1
w 1
w 1
w 4
w
4
w 1
w 2
w 3
w 1
w

Table (4)
With KU-function, f: V → V, f(x) = x.
Example 3.6. Let V = {101000, 110000,101100, 111111, 100000, 111010} be a binary block code. Using the
lexicographic order, the code V can be written V = {100000, 110000, 101000, 101100,111010, 111111}
= },,,,,{ 654321
wwwwww .With the following graph:
Fig. (2): A graph of code with 6 vertices and 6 edges
In figure (2): },,,,,{ 654321
wwwwww are the set of vertices and
{ },{},,{},,{},,{},,{ 4365315221
wwwwwwwwww } are the set of edges.
By using the previous theorem we define the partial order  on V then we get ,1 i
WW  i∈ {2, 3, 4, 5, 6},
2
W Can’t be compared with 3
W , 2
W can’t be compared with 4
W , 52
WW  , 62
WW  , 43
WW  ,
53
WW  , 63
WW  , 4
W , and 64
WW  .The operation [*] on V is defined by the
following table:
Table (5)
Then, V with the operation [*] is a KU-algebra. The same binary block code V can be obtained from a KU-
algebra ),,( B
Table (6)
θ a b c
Θ θ a b c
a θ θ b c
b θ θ θ c
c θ a b θ

Example 3.7. Let V = {1010, 1100, 1011, 1000} be a binary block code, using the lexicographic order, the code
V can be written V = {1000, 1100, 1010, 1011} = },,,{ 4321
wwww . With the following graph
Fig. (3): A graph with 4 vertices and 3 edges.
In figure (3): },,,{ 4321
wwww are the set of vertices and { },{},,{},,{ 433121
wwwwww } are the set of
edges.
By using the previous theorem we define the partial order ≤ on V, then we get i
WW 1
,i∈{2,3,4}, 2
W , 2
W ,and 43
WW  .The operation [*] on
V is defined by the following table:
Table (7)
Then, V with the operation [*] is a KU-algebra.
The same binary block code V can be obtained from a KU-algebra ),,( C
Table (8)
Proposition 3.8. Suppose that })1,0({)( ,
},...,2,1{
},.....,2,1{, mn
mj
niji
McC 


is a matrix with rows lexicographic
ordered in the ascending sense, so there is a matrix mnqMsS qqjiji
 
}),1,0({)( },....,2,1{,,
, such that
S is a lower triangular matrix, with },...,2,1{,1 qiS ii
 and C becomes a sub matrix of the matrix S.
Proof. Suppose that we add in the right side of the matrix C (from the left to the right) the new rows of the
form 
nnn
01.....00,.......,00...01,00.....10 , so we have a new matrix S with n + m columns and n rows. Suppose
that we add in the top of the matrix S the following n rows:
1......000,,.........guatda.com/cmx.p00...010.....00,00...100...00
111

 mnmnmn
.We get the required matrix C.
1
w 2
w 3
w 4
w
1
w 1
w 2
w 3
w 4
w
2
w 1
w 1
w 3
w 4
w
3
w 1
w 2
w 1
w 4
w
4
w 1
w 2
w 1
w 1
w
θ a B c
θ θ a B C
a θ θ B c
b θ a Θ c
c θ a Θ θ

Theorem 3.9. Let V be a binary block code with n codewords of length m , n≠m, or a block-code with n
codewords of length n such that }1,0{......;......1 inikinik
xxxx is not in V, or a block-code with n
codewords of length n such that the matrix V
M is not lower triangular. Then Through the previous notations,
we have a natural number q ≥ max {m, n}, a set A with m elements and a KU function f: A →Cq such that the
obtained block code Cn
V contains the block code V with 1s as a first digit in its codewords .
Proof. Suppose that V= },,,{ 4321
wwww , be a binary block code, with codewords of length m. We consider
the codewords n
www ,........,, 21
lexicographic ordered nlexlexlex
www  .........21
. Suppose that
})1,0({,mn
MM  be the associated matrix of V with the rows n
www ,........,, 21
in this order. By using
Proposition 3.8, we lengthen the matrix M to a square matrix M  ∈ mnqM q
}),1,0({ , such that
},....,2,1{,,
)( qjiji
mM 
 is a lower triangular matrix with ,1ii
m , for all i∈{1,2,...,q}.If the first column of
the matrix M  is not 11….1(q-times), then we insert the column 11…..1(q+1 times) as a first column and the
row 10….0 (number of zeroes =q-times) as a first row. Applying Theorem 3.2 for the matrix M  , we obtain a
KU-algebra },.....,,{ 21 qq
xxxC  ,with 1
x = θ the zero of the algebra q
C and a binary block code cq
V .
Supposing that the columns of the matrix M have in the new matrix M  with 1s as a first digit, so
.},......,,{ 21 qmjjj
CxxxA  The KU-function },.....,2,1{,)(,: mixxfCAf ijijq
 , determines
the binary block-code cq
V such that the code cq
V contains the block code V with 1s as a first digit in its
codewords.
Example 3.10. Let V = {01101, 00001, 00101, 01111} be a binary block code. By using the lexicographic order,
the code V can be written V = {00001, 00101, 01101, 01111} = },,,{ 4321
wwww . We organize the codewords
in the associated matrix M, such that }).1,0({5,4
MM V
 we get,















11110
10110
10100
10000
V
M
By using proposition 3.8, we create a lower triangular matrix.















100011110
010010110
001010100
000110000
F and





























100011110
010010110
001010100
000110000
000010000
000001000
000000100
000000010
000000001
S

The rows of the matrix S doesn’t begin with 1.by using proposition 3.8, we add 11…..1(10-times) as a first
column and the row 10….0 (number of zeroes is 9-times) as it don’t exist in the first row of the matrix S. so we
get the following:

































1000111101
0100101101
0010101001
0001100001
0000100001
0000010001
0000001001
0000000101
0000000011
0000000001
S
The binary block code }.,,.........,{ 1021
wwwW  , whose codewords are the rows of the matrix S  ,
determines a KU-algebra (X,*, 1
w ).
Suppose that },,,,,{ 654321
wwwwwwA  and ii
wwfXAf  )(,: ,i∈{1,2,3,4,5,6} be a KU –
function which determines the binary block code U= {100001, 110000, 101000, 100100, 100010, 100001,
100001,100101,101101,101111}.We have the code V included in the code U but it contains 1s as a first digit .
IV. Relationship between the ordered relation on KU-algebra and partial ordered set
Definition 4.1. Suppose that ),( P is a partially ordered set. For Pq  , we define a mapping
}1,0{: PPq
such that for each Pb  , we have 1)( bPq
if and only if bq  , Using this map, a
codeword nx
xxxv ....21
 of a binary block-code V can be determined as follow: Jx i
 if and only if
}.1,0{,)(  jandSiforjiPx
From a given partially ordered set we catch binary block codes as showing in the following examples:
Example 4.2. Suppose that P = {0, 1, 2, 3} is a set with a partial order over P as presented in the following
figure (4)
Fig. (4): partial order ( ,P )
We catch the following table by using definition 4.1:
p
P 0 1 2 3
0 1 0 0 0
1 1 1 0 0
2 1 0 1 0
3 1 0 1 1
Table (9)

From table (9) we get the following code }1011,1010,1100,1000{1
V , and we get following figure (5) from
code 1
V .
Fig. (5): order relation ),( 1 C
V 
Example 4.3. Suppose that P= {0, 1, 2, 3, 4, 5} is a set with a partial order over P as presented in the following
figure (6)
Fig. (6): partial order ( ,P )
We catch the following table by using definition4.1:
p
P 0 1 2 3 4 5
0 1 0 0 0 0 0
1 1 1 0 0 0 0
2 1 1 1 0 0 0
3 1 1 0 1 0 0
4 1 0 0 0 1 0
5 1 0 0 0 1 1
Table (10)
From table (10) we get the following code }100011,100010,110100,111000,110000,100000{2
V , and
we get following figure (7) from code 2
V .
Fig. (7): order relation ( ),( 2 C
V  )
Now, we generate binary block codes from KU-algebras by using definition 2.16 through the following
examples.
Example 4.4.Suppose that X = {0, 1, 2, 3} is a KU-algebra and we represent the order on X as shown in
figure (5).

* 0 1 2 3
0 0 1 2 3
1 0 0 2 3
2 0 1 0 3
3 0 1 0 0
Table (11)
Suppose that XXA :
~
is a KU-function on X given by 








3210
3210~
A . So, we have the following
table.
x
A
~ 0 1 2 3
0
~
A
1 0 0 0
1
~
A
1 1 0 0
2
~
A
1 0 1 0
3
~
A
1 0 1 1
Table (12)
We observe in table (12) the binary block code }1011,1010,1100,1000{3
V that equal the code 1
V in
example 4.2.
Example 4.5. Suppose that X = {0, 1, 2, 3, 4, 5} is a KU-algebra and we represent the order on X as shown in
figure (7).
* 0 1 2 3 4 5
0 0 1 2 3 4 5
1 0 0 2 3 4 5
2 0 0 0 3 4 5
3 0 0 2 0 4 5
4 0 1 2 3 0 5
5 0 1 2 3 0 0
Table (13)
Suppose that XXA :
~
is a KU-function on X given by 








3210
3210~
A . So, we have the following
table.
Table (14)
Table (14)
x
A
~ 0 1 2 3 4 5
0
~
A
1 0 0 0 0 0
1
~
A
1 1 0 0 0 0
2
~
A
1 1 1 0 0 0
3
~
A
1 1 0 1 0 0
3
~
A
1 0 0 0 1 0
3
~
A
1 0 0 0 1 1

We observe in table (12) the binary block code }1011,1010,1100,1000{3
V that equal the code 1
V in
example 4.3.
We define a KU-algebra structure on a poset with 0 element and we have a code in Example 4.2 similar to the
code in Example 4.4, also we have a code in Example 4.3 similar to the code in Example 4.5. The clear is that
we use the order of KU-algebra only, not its properties. From the previous examples, we deduce that there is a
one to one correspondence between the ordering relation  and order relation c
 .
Proposition 4.6. There is a one to one correspondence between the ordered relation on KU-algebra and partial
ordered set.
References
[1] Bollobás, B. (1979). Graph Theory: An Introductory CourseSpringer-Verlag.New York/Berlin.
[2] Bronson, R. (1989). Schaum's outline of theory and problems of matrix operations.
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[4] Dudek, W. A., & Zhang, X. (1992). On proper BCC-algebras. Bull. Inst. Math. Academia Sinica, 20, 137-150.
[5] Flaut, C. (2015). BCK-algebras arising from block codes. Journal of Intelligent & Fuzzy Systems, 28(4), 1829-1833.
[6] Imai, Y., & Iséki, K. (1966). On axiom systems of propositional calculi. XIV.Proceedings of the Japan Academy, 42(1), 19-22.
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pp. 3139-3149).
[10] Mostafa, S. M., Youssef, B. A., & Jad, H. A. (2015). Coding theory applied to KU-algebras. arXiv preprint arXiv:1505.04396.
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Letters, 2015, Article-ID.
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Forum (Vol. 6, No. 64, pp. 3151-3159).
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