![I.1.1 Balance 7
− − − − − − −−
− − − + − + ++
− + − − + − ++
− + + − − + −+
− + + + − − +−
− − + + + − −+
− + − + + + −−
− − + − + + +−
Place a − in the cell of row x and column
y when symbol x appears in block y, and
a + otherwise. Finally add an initial row
and an initial column consisting entirely
of − to get the array at the left. Invert +
and − to get the array on the right.
+ + + + + + ++
+ + + − + − −−
+ − + + − + −−
+ − − + + − +−
+ − − − + + −+
+ + − − − + +−
+ − + − − − ++
+ + − + − − −+
1.21 Remark Of course, the balance observed previously repeats in the matrix on the left in
Example 1.20. In this presentation, however, the balance is seen in a new light. Taking
+ to be +1 and − to be −1, it is possible for two rows to have inner product between
−8 and +8; yet here every pair of distinct rows have inner product exactly equal to
0 (and indeed, so do any two distinct columns). This property over {±1} defines a
Hadamard matrix; see §V.1. Such matrices have the remarkable property that they
maximize the matrix determinant over all n×n {±1}-matrices; see §V.3. Generalizing
to arithmetic over algebras other than ({±1}, ·) yields classes of orthogonal matrices
and designs (§V.2).
The matrix on the right shares the same properties, being the negative of the one
on the left. But a surprise is in store. Juxtapose the two horizontally, and delete the
two constant columns to form an 8 × 14 matrix. Every two distinct columns of the
result have either 0 or 4 identical entries. For each column form a set that contains
the indices of the + entries in that column; this yields 14 blocks each of size 4, on a
set of 8 elements. Every element occurs in 7 blocks, every 2-subset of elements in 6
blocks, and every 3-subset of elements in exactly one block. This is a Steiner system
and a 3-design (a Hadamard 3-design).
1.22 Example In Example 1.20, delete the constant row and constant column, and treat
the remaining 7 rows as {±1}-vectors. These vectors are:
[--+-+++] [+--+-++] [++--+-+] [+++--+-] [-+++--+] [+-+++--] [-+-+++-]
1.23 Remark In communicating between a sender and a receiver, synchronization is a ma-
jor issue, as demonstrated by the message ...EATEATEATEATEATEATEATEA..., which
might say EAT, ATE, or TEA depending on your timing. Knowing the message, how-
ever, enables one to determine the timing. Suppose then that a sender repeatedly sent
[+-+++--]. If the receiver is “in phase,” it multiplies the expected vector with the
received one, and computes 7. If “out of phase,” it instead computes −1 no matter
what the nonzero phase shift is. This autocorrelation can be used to synchronize, and
the balance plays a practical purpose; see §V.9 and §V.7.
1.24 Example An error-correcting code.
Replace + by 1 and − by 0 in the two Hadamard ma-
trices of Example 1.20, and treat the 16 rows of the
resulting matrices as binary vectors of length 8.
[00000000] [00010111]
[01001011] [01100101]
[01110010] [00111001]
[01011100] [00101110]
[11111111] [11101000]
[10110100] [10011010]
[10001101] [11000110]
[10100011] [11010001]
1.25 Remark The sixteen binary vectors of Example 1.24 have the property that every
two distinct vectors differ in exactly 4 or in all 8 positions. A sender and receiver
can agree that in order to communicate sixteen information symbols, they are to use
these 16 “codewords.” If a single bit error is made in transmission, there is exactly
one codeword that could have been intended; if two bit errors are made, the receiver
knows that an error occurred but does not know which two bit errors account for the
incorrect received word. This is a 1-error-correcting code and a 2-error-detecting code;
see §VII.1.](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-37-320.jpg)
![12 Design Theory: Antiquity to 1950 I.2
The literature on latin squares goes back at least 300 years to the monograph
Koo-Soo-Ryak by Choi Seok-Jeong (1646–1715); he uses orthogonal latin squares of
order 9 to construct a magic square and notes that he cannot find orthogonal latin
squares of order 10. It is unlikely that this was the first appearance of latin squares.
Ahrens [68] remarks that latin square amulets go back to medieval Islam (c1200), and
a magic square of al-Buni, c 1200, indicates knowledge of two 4 × 4 orthogonal latin
squares. In 1723, a new edition of Ozanam’s four-volume treatise [1712] presented a
card puzzle that is equivalent to finding two orthogonal latin squares of order 4. Then
in 1776, Euler presented a paper (De Quadratis Magicis) to the Academy of Sciences
in St. Petersburg in which he again constructed magic squares of orders 3, 4, and
5 from orthogonal latin squares. He posed the question for order 6, now known as
Euler’s 36 Officers Problem. Euler was unable to find a solution and wrote a more
extensive paper [798] in 1779/1782. He conjectured that no solution exists for order
6. Indeed he conjectured further that there exist orthogonal latin squares of all orders
n except when n ≡ 2 (mod 4):
et je n’ai pas hésité d’en conclure qu’on ne saurait produire aucun quarré complet de 36
cases, et que la même impossibilité s’étende aux cas de n = 10, n = 14 et en général à tous
les nombres impairement pairs.
No progress was to be made on this problem for over a century, although it was not
neglected by mathematicians of the day. Indeed, Gauss and Schumacher (see [862])
corresponded in 1842 about work of Clausen establishing the impossibility when n = 6
and conjecturing (independently of Euler) the impossibility when n ≡ 2 (mod 4),
which work was apparently not published. In 1900, by an exhaustive search, Tarry
[2004] showed that none exists for n = 6, and in 1934 the statisticians Fisher and
Yates [820] gave a simpler proof. Petersen [1731] in 1901 and Wernicke [2129] in
1910 published fallacious proofs of Euler’s conjecture for n ≡ 2 (mod 4). Then in
1922 MacNeish [1499] published another, after pointing out the error in Wernicke’s
approach! In 1942 Mann [1520] described the construction of orthogonal latin squares
using orthomorphisms. The falsity of the general Euler conjecture was finally estab-
lished in 1958 when Bose and Shrikhande [308] constructed two orthogonal squares of
order 22; and then in 1960 Bose, Shrikhande, and Parker [309] established that two
orthogonal latin squares of order n exist for all n ≡ 2 (mod 4), other than 2 and 6.
The study of block designs can be traced back to 1835, when Plücker [1754], in a
study of algebraic curves, encountered a Steiner triple system of order 9, and claimed
that an STS(m) could exist only when m ≡ 3 (mod 6). In 1839 he correctly revised
this condition to m ≡ 1, 3 (mod 6) [1755]. Plücker’s discovery that every nonsingular
cubic has 9 points of inflection, defining 12 lines each of 3 points, and with each pair
of points on a line, illustrates the early connection between designs and geometry.
Another discovery of Plücker concerned the double tangents of fourth order curves,
namely, a design of 819 4-element subsets of a 28-element set such that every 3-element
subset is in exactly one of the 4-element subsets, a 3-(28, 4, 1) design. Plücker raised
the question of constructing in general a t-(m, t + 1, 1) design.
In England, a prize question of Woolhouse [2167] in the Lady’s and Gentleman’s
Diary of 1844 asked:
determine the number of combinations that can be made out of n symbols, p symbols in
each; with this limitation, that no combination of q symbols, which may appear in any one
of them shall be repeated in any other.
In 1847 Kirkman [1300] dealt with the existence of such a system in the case p = 3
and q = 2, constructing STS(m) for all m ≡ 1, 3 (mod 6). The next case to be solved,
p = 4 and q = 3, was done more than a century later by Hanani [1035] in 1960. The
reason the triad systems (as Kirkman called them) are not now called Kirkman triple](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-42-320.jpg)
![I.2 Design Theory: Antiquity to 1950 13
systems but rather Steiner triple systems lies in the fact that Steiner [1956], apparently
unaware of Kirkman’s work, asked about their existence in 1853; subsequently Reiss
[1799] (again) settled their existence, and later Witt [2159] named the systems after
Steiner. A particular irony is that Steiner had actually asked a related but different
question, which was taken up by Bussey in 1914 [407]. The name Kirkman has instead
come to be attached to resolvable Steiner systems, on account of Kirkman’s famous
15 schoolgirls problem [1303]: fifteen young ladies in a school walk out three abreast
for seven days in succession; it is required to arrange them daily, so that no two shall
walk twice abreast.
The first solution to the schoolgirls problem to appear in print was by Cayley [442]
in 1850 followed in the same year by another by Kirkman [1302] himself. Kirkman’s so-
lution involved an ingenious combination of an STS(7) and a (noncyclic) Room square
of side 7. In 1863, Cayley [443] pointed out that his solution could also be presented
in this way. Cayley’s and Kirkman’s solutions are nonisomorphic as Kirkman triple
systems but isomorphic as Steiner triple systems, being the two different resolutions
into parallel classes of the point-line design of the projective geometry PG(3, 2). In
1860 Peirce [1726] found all three solutions of the 15 schoolgirls problem having an
automorphism of order 7, and in 1897 Davis [634] published an elegant geometric solu-
tion based on representing the girls as the 8 corners, the 6 midpoints of the faces, and
the centre of a cube. In 1912, Eckenstein [769] published a bibliography of Kirkman’s
schoolgirl problem, consisting of 48 papers. The 7 nonisomorphic solutions of the 15
schoolgirls problem were enumerated by Mulder [1644] and Cole [593]. The first pub-
lished solution of the existence of a Kirkman triple system of order m, KTS(m), for
all m ≡ 3 (mod 6) was by Ray-Chaudhuri and Wilson [1781] in 1971. The problem
had in fact already been solved by the Chinese mathematician Lu Jiaxi at least eight
years previously, but the solution had remained unpublished because of the political
upheavals of the time. Lu also solved the corresponding problem for blocks of size 4;
these results were eventually published in his collected works [1484] in 1990.
Prior claim to the authorship of the 15 schoolgirls problem was made by Sylvester
[1996] in 1861, hotly refuted by both Kirkman [1305] and Woolhouse [2168]. But
Sylvester raised further questions about such systems, for example: can all 455 triples
from a 15-element set be arranged into 13 disjoint KTS(15)s, thereby allowing a walk
for each of the 13 weeks of a school term, without any three girls walking together
twice? The problem was finally solved in 1974 by Denniston [690] although both
Kirkman and Sylvester had solutions for the corresponding problem with 9 instead of
15 (indeed they had the first example of a large set of designs). Surely neither Sylvester
nor Denniston would have anticipated that Denniston’s solution would form the basis
in 2005 of a musical score, Kirkman’s Ladies [1212], about which the composer states:
The music seems to come from some point where precise organization meets near chaos.
The problem of the existence of large sets of KTS(n) in the general case of n ≡ 3
(mod 6) is still unsolved. Sylvester also asked about the existence of a large set of 13
disjoint STS(15)s. In this case, the general problem of large sets of STS(n) is solved
for all n ≡ 1, 3 (mod 6), n 6= 7, mainly by work of Lu [1482, 1483] but also Teirlinck
[2015]. Cayley [442] had shown that the maximum number of disjoint STS(7)s on the
same set is 2. In 1917, Bays [169] determined that there are precisely 2 nonisomorphic
large sets of STS(9).
Kirkman’s solution of the 15 schoolgirls problem was extended brilliantly in 1852–
3 by Anstice [103, 104]; he generalised the case where n = 15 to any integer of the
form n = 2p + 1, where p is prime, p ≡ 1 (mod 6). For the first time making use of
primitive roots in the construction of designs (a method basic to Bose’s 1939 paper)
and introducing the method of difference families, Anstice constructed an infinite class](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-43-320.jpg)
![14 Design Theory: Antiquity to 1950 I.2
of cyclic Room squares, cyclic Steiner triple systems, and 2-rotational Kirkman triple
systems. His starter-adder approach to the construction of Room squares includes the
Mullin–Nemeth starters [1653] as a special case.
In 1857 Kirkman [1304] used difference sets (in the equivalent formulation of perfect
partitions) to construct cyclic finite projective planes of orders 2, 3, 4, 5, and 8,
introducing at the same time the concept of a multiplier of a difference set. He also
wrote that he thought that a solution for 6 is “improbable,” for 7 is “very likely”
(because 7 is prime) and that he did not see why one should not be discovered for
9. He did not comment on order 10. A few years earlier, Kirkman [1301] had shown
how to construct affine planes of prime order, and, hence, essentially by adding points
at infinity, finite projective planes of all prime orders (although he did not use the
geometric terminology). This paper also showed how to construct certain families of
pairwise balanced designs. Such designs were to prove invaluable in later combinatorial
constructions; indeed, in the same paper Kirkman himself used them to solve the
schoolgirls problem for 5 · 3m+1
symbols. Kirkman’s work on combinatorial designs is
indeed seminal. To quote Biggs [269]:
Kirkman has established an incontestable claim to be regarded as the founding father of
the theory of designs. Among his contemporaries, only Sylvester attempted anything com-
parable, and his papers on Tactic seem to be more concerned with advancing his claims to
have discovered the subject than with advancing the subject itself. Not until the Tactical
Memoranda of E. H. Moore in 1896 is there another contribution to rival Kirkman’s.
In 1867, Sylvester [1997] investigated a tiling problem and constructed what he
called anallagmatic pavements, which are equivalent to Hadamard matrices. He showed
how to construct such of order 2n from a solution of order n; twenty-six years later
Hadamard [1003] showed that Hadamard matrices give the largest possible determi-
nant for a matrix whose entries are bounded by 1. Hadamard showed that the order
of such a matrix had to be 1, 2, or a multiple of 4, and he constructed matrices of
orders 12 and 20. In 1898 Scarpis [1846] showed how to construct Hadamard matrices
of order 2k
· p(p + 1) whenever p is a prime for which a Hadamard matrix of order
p + 1 exists. In 1933 Paley [1713] used squares in Fq to construct Hadamard matrices
of order q + 1 when q ≡ 3 (mod 4) and of order 2(q + 1) when q ≡ 1 (mod 4). He
also showed that if m = 2n
, then the 2m
(±1)-sequences of length m can be parti-
tioned in such a way as to form the rows of 2m−n
Hadamard matrices. In a footnote
Paley acknowledged that some of his results had been announced by Gilman in the
1931 Bulletin of the AMS, but without proof. In the same journal issue in 1933,
Todd [2036] pointed out the connection between Hadamard matrices and Hadamard
designs; designs with these parameters were relevant to work of Coxeter (also in the
same journal issue) on regular compound polytopes! At this time the matrices were
not yet called Hadamard matrices; Paley called them U-matrices. Williamson [2142]
first called them Hadamard matrices in 1944. The existence of Hadamard matrices
for all admissible orders remains open.
Cayley introduced the word tactic for the general area of designs. The word config-
uration, used nowadays more generally, was given a specific meaning in 1876 by Reye
[1800] in terms of geometrical structures. Essentially, a configuration was defined to
be a system of v points and b lines with k points on each line and r lines through
each point, with at most one line through any two points (and hence any two lines
intersecting in at most one point). If v = b (and therefore k = r), the configuration is
symmetric and denoted by vk; thus, for example, the Fano plane is 73, the Pappus con-
figuration is 93, and the Desargues configuration is 103. In fact, the first appearance
of the Petersen graph is not in Petersen’s paper of 1891 but in 1886 by Kempe [1277]
as the graph of the Desargues configuration. In 1881, Kantor [1250] showed that there
exist one 83, three 93, and ten 103 configurations. Six years later, Martinetti [1528]](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-44-320.jpg)
![I.2 Design Theory: Antiquity to 1950 15
showed there were thirty-one 113 configurations. In the following years, Martinetti
constructed symmetric 2-configurations (two points are connected by at most 2 lines),
with parameters (74)2, (84)2, and (94)2 as well as the biplanes (115)2 and (166)2. The
biplane (115)2, also a Hadamard design 2-(11, 5, 2), had previously been constructed
by Kirkman [1305] in 1862. Symmetric configurations v4 were first studied in 1913
by Merlin [1593]. In 1942, Levi [1442] unified the treatment of configurations with
questions in algebra, geometry, and design theory.
Interest in counting configurations is reflected in the enumerative work on Steiner
systems. It was known early on that there are unique STS(7) and STS(9). In 1897,
Zulauf [2219] showed that the known STS(13)s fall into two isomorphism classes.
Using the connection between STS(13)s and symmetric configurations 103, in 1899 De
Pasquale [659] determined that only two isomorphism classes are possible. The same
result was also obtained two years later by Brunel. The enumeration of nonisomorphic
STS(15)s was first done by Cole, Cummings, and White. In 1913, both Cole and
White published papers introducing ideas for enumeration and a year later Cummings
classified all 23 STS(15)s which have a subsystem of order 7. Then in 1919, White,
Cole, and Cummings [2135] in a remarkable memoir, succeeded in determining that
there are precisely 80 nonisomorphic STS(15)s. In 1940, Fisher [819], unaware of this
catalogue, also generated STS(15)s; he used trades on Pasch configurations to find 79
of the 80 systems. The veracity of White, Cole, and Cummings’ work was confirmed in
1955 by Hall and Swift [1020] in one of the first cases in which digital computers were
used to catalogue combinatorial designs. The number of nonisomorphic STS(19)s
was published only in 2004 [1268], perhaps understandably as it is 11,084,874,829.
Enumeration of latin squares has a more complex history (see [1573]); early results
are by Euler [798] in 1782, Cayley [446] and Frolov [834] in 1890, Tarry [2004] in 1900,
MacMahon [1498] in 1915, Schönhardt [1853] in 1930, Fisher and Yates [820] in 1934,
and Sade [1836] and Saxena [1844] in 1951.
By the end of the 19th century there was no formal body of work called graph
theory, but some graph theoretic results were in existence in the language of config-
urations or designs. One-factorizations of K2n are resolvable 2-(2n, 2, 1) designs, and
the “classical” method of construction appeared in Lucas’ 1883 book [1488]; the con-
struction was credited to Walecki and can be described in terms of chords of a circle
or in terms of a difference family now known as a patterned starter. Remarkably, in
1847 Kirkman [1300] already gave a construction based on a lexicographic method
(in fact the greedy algorithm), which gives a schedule isomorphic to that of Lucas–
Walecki. Lucas’ book also describes the construction of round-dances (or hamiltonian
decompositions of the complete graph) by means of what we now call a terrace. Enu-
meration results for nonisomorphic 1-factorizations of K2n span nearly a century. In
1896, Moore [1626] reported that there are 6 distinct 1-factorizations of K6 that fall
into a single isomorphism class. The 6 nonisomorphic 1-factorizations of K8 appear
in Dickson and Safford [703]. Gelling [887] showed that the number of nonisomorphic
1-factorizations of K10 is 396, and Dinitz, Garnick, and McKay [715] showed that
there are 526,915,620 nonisomorphic 1-factorizations of K12.
In 1891, Netto [1670], unaware of Kirkman’s work, gave four constructions for
Steiner triple systems: (i) an STS(2n+1) from an STS(n), (already used by Kirkman),
(ii) an STS(mn) from an STS(m) and an STS(n), (iii) an STS(p) where p is a prime of
the form 6m+1, and (iv) an STS(3p) where p is of the form 6m+5. The construction
given by Netto for (iii) involves primitive roots, but is not the construction usually
associated with his name. These constructions enabled Netto to construct STS(n)
for all admissible n < 100 except for n ∈ {25, 85}. In 1893 Moore [1624] completed
Netto’s proof, giving a construction of an STS(w + u(v − w)) from an STS(u) and an
STS(v) with an STS(w) subsystem. In [1624] he also proved that for all admissible](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-45-320.jpg)
![16 Design Theory: Antiquity to 1950 I.2
v > 13, there exist at least two nonisomorphic STS(v). In 1895 and 1899, Moore
[1625, 1627] determined the automorphism groups of the unique STS(7) and 3-(8, 4, 1)
designs and studied the intersection properties of the 30 realizations on the same base
set of both systems.
Undoubtedly the most far-reaching paper by Moore is his “Tactical Memoranda
I–III” [1626]. This was published in 1896 in the American Journal of Mathematics
(founded in 1878 by Sylvester during his seven-year stay in the U.S. as professor at
Johns Hopkins University). It contains many jewels hidden in 40 pages of difficult
and often bewildering terminology and notation. After a brief introductory section
(memorandum I), memorandum II introduces a plethora of different types of designs,
including with order and/or with repeated block elements, the first occurrence of both
of these concepts. Here we find the construction using finite fields of a complete set
of q − 1 mutually orthogonal latin squares (MOLS) of order q whenever q is a prime
power (a result rediscovered much later by Bose [300] in 1938 and by Stevens [1961]
in 1939 independently). One also finds the theorem, later rediscovered by MacNeish
[1499], that if there exist t MOLS of order m and of order n, then there exist t
MOLS of order mn. Further, it contains the construction of many 1-rotational designs
including Kirkman triple systems and work on orthogonal arrays. Memorandum III
contains the first general results on whist tournaments and triple-whist tournaments.
Whist tournaments Wh(4n) had been presented by Mitchell [1618] in 1891 for several
small values of n; Moore showed how to construct several infinite families of such
designs. These are resolvable 2-(4n, 4, 3) designs with extra properties, and are the
first examples of nested designs to appear in the literature. Further in this section,
Moore points out the relationship between whist tournaments Wh(4n) and resolvable
2-(4n, 4, 1) designs and gives a cyclic construction of such designs in the case where
4n = 3p + 1, where p is a prime of the form 4m + 1. At this time much work
was being done on finite geometries and tactical configurations. For example, Fano
[807] described a number of finite geometries in 1892, and in 1906 Veblen and Bussey
[2097] used finite projective geometries to construct many designs, in particular finite
projective planes of all prime power orders.
Triple systems were also studied by Heffter whose approach to the subject stemmed
from his study of triangulations of complete graphs on orientable surfaces, relating
these to twofold triple systems. In 1891 he gave a particularly simple construction
for twofold triple systems of order 12s + 7 [1076]. Although Kirkman, nearly 40 years
earlier, had pointed to the existence of an indecomposable 2-(7, 3, 3) design without
repeated blocks, this seems to be the first known infinite class of designs with λ > 1.
In 1896, Heffter [1077] introduced his famous first difference problem, in relation to
the construction of cyclic STS(6s + 1), and a year later both the first and second
difference problems appeared [1078].
Heffter’s difference problems were eventually solved in 1939 by Peltesohn [1727],
and later a particularly elegant approach by Skolem [1924, 1925] was given in 1957–
58. Skolem’s interest in Steiner triple systems was longstanding. In 1927 he wrote
notes for the second edition of Netto’s book [1671], whose first edition appeared in
1901; there he constructed STS(v) for all products of primes of the form 6n + 1 and
showed that if v = 6s + 1 then there are at least 2s such STS(v). Then in [1924], he
introduced the idea of a pure Skolem sequence of order n and proved that these exist
if and only if n ≡ 0, 1 (mod 4). In [1925] he extended this idea to that of a hooked
Skolem sequence, the existence of which for all admissible n, along with that of pure
Skolem sequences, would constitute a complete solution to Heffter’s first difference
problem. O’Keefe [1695] proved that a hooked Skolem sequence exists if and only if
n ≡ 2, 3 (mod 4).
In 1923–34, Bays [170, 171] undertook a systematic study of group actions on](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-46-320.jpg)
![I.2 Design Theory: Antiquity to 1950 17
cyclic STS, and also in [172] on SQS; this culminated in a remarkable theorem, the
Bays–Lambossy theorem [1382] demonstrating that isomorphism of cyclic structures
of prime orders is the same as multiplier equivalence.
The period between the two world wars saw a major step forward in the study
of statistical experimental design. Already by 1788, de Palluel [618] had used a 4 ×
4 latin square to define an experimental layout for wintering sheep (see [1985] for
a modern treatment of his work). In 1924 Knut Vik [2100] described the use of
certain latin squares in field experiments. But in 1926 [818], Fisher, at Rothamsted
Experimental Station, indicated how orthogonal latin squares could be used in the
construction of experiments. This led to detailed work in listing and enumerating all
latin squares of small orders; in particular, in 1934 Fisher and Yates [820] (Fisher’s
successor at Rothamsted when he left to take up a chair in London) enumerated
all 6 × 6 latin squares and established that no two were orthogonal. Then, in a
1935 paper delivered to the Royal Statistical Society, Yates [2180] drew attention
to the importance of balanced block designs for statistical design. In [2181] Yates
called them symmetrical incomplete randomized blocks, the present terminology of
balanced incomplete block designs being introduced by Bose in 1939; but the present
use of v, b, r, k, and λ goes back to Yates (except that Yates originally used t for
treatment instead of v for variety). It was also in 1935 that Yates [2179] gave
the first formal treatment of the factorial designs proposed by Fisher in 1926. In
1938, Fisher and Yates [821] published their important book Statistical Tables for
Biological, Agricultural and Medical Research which, along with much statistical data,
presented what was known about latin squares of small order, including complete sets
of orthogonal latin squares of orders 3, 4, 5, 7, 8, and 9, as well as tables of balanced
incomplete block designs with replication number r up to 10. Some gaps would be filled
by Bose the following year, and some gaps would be ruled out by Fisher’s inequality.
The years 1938–39 saw the publication of several important papers. Peltesohn
[1727] gave a complete solution to Heffter’s difference problems and hence established
the existence of cyclic Steiner triple systems for all admissible orders with the exception
of v = 9. Singer [1920] used hyperplanes in PG(2, q) to construct cyclic difference sets,
thereby establishing the existence of cyclic finite projective planes (a few of which had
been found by Kirkman) for all prime power orders. Singer’s cyclic difference sets gave
the first important family of difference sets apart from those implicitly in the work of
Paley, although many examples of difference families had been used by Anstice and
Netto. In 1942 Bose [302] gave an affine analogue of Singer’s theorem. Here he gave the
first examples of relative difference sets and suggested the notion of a group divisible
design (GDD). Chowla [501] constructed difference sets using biquadratic residues in
1944. In 1947, Hall [1012] published an important paper on cyclic projective planes.
Difference sets were studied in depth after World War II; the important multiplier
theorem of Hall and Ryser [1019] dates from 1951.
In 1938, there was also the remarkable paper by Witt [2159] on Steiner systems,
detailing the existence of several Steiner systems with t = 3, including various families
obtained geometrically and the Steiner systems S(3, 4, 26) and S(3, 4, 34) constructed
by Fitting [822] in 1914. But of greater interest, the paper also includes the systems
S(5, 8, 24) and S(5, 6, 12) with the Mathieu groups M24 and M12, respectively, as their
automorphism groups. Witt gives a construction of these systems as a single orbit of
a starter block under the action of the groups PSL2(23) and PSL2(11), respectively,
and proofs of the uniqueness of the designs as well as of their derived subsystems. He
also gives a proof of the uniqueness of the system S(3, 5, 17) and of the nonexistence
of a system S(4, 6, 18).
The results on the Mathieu systems had already appeared in 1931 by Carmichael
[437, 438]. However in 1868 in the Educational Times, Lea [1418] both proposed the](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-47-320.jpg)
![18 Design Theory: Antiquity to 1950 I.2
problem and gave a solution to constructing the system S(4, 5, 11). The next year
another solution appeared, this time by none other than Kirkman [1306], who went
on to try to construct a system S(4, 5, 15). The fact that this system does not exist
was not shown until 1972 by Mendelsohn and Hung [1588]. Because it is relatively
easy to construct the system S(5, 6, 12) from the S(4, 5, 11), it is perhaps a surprise
that Kirkman did not do so. Instead, this was left to Barrau [161] in 1908 who,
using purely combinatorial methods, proved the existence and uniqueness of both the
systems S(4, 5, 11) and S(5, 6, 12).
The work of Fisher and Yates created growing interest in designs among statisti-
cians, and in 1939 Bose [301], in Calcutta, published a major paper on the subject.
To quote Colbourn and Rosa [578]:
Sixty years on, the paper of Bose forms a watershed. Before then, while combinatorial design
theory arose with some frequency in other researches, it was an area without a basic theme,
and without substantial application. After the paper of Bose, the themes of the area that
recur today were clarified, and the importance of constructing designs went far beyond either
recreational interest, or as a secondary tactical problem in a larger algebraic or geometric
study.
Bose presents the study of balanced incomplete block designs as a coherent theory,
using finite fields, finite projective geometries, and difference methods to create many
families of designs that contain many old and many new examples. There is much
material on designs with λ > 1. The use of difference families, both pure and mixed,
carried on the work of Anstice of which he was unaware. Bose’s paper appeared
in volume 9 of the Annals of Eugenics. It is remarkable what combinatorial design
theory appeared in that volume of 1939. Stevens [1961] discusses the existence of
complete sets of orthogonal latin squares; Savur [1843] gives (yet) another construction
of STSs; and Norton [1686] discusses 7×7 latin and graeco-latin squares. Stevens and
Norton had presented their work at the 1938 meeting of the British Association in
Cambridge, where Youden [2195] introduced the experimental designs subsequently
known as Youden squares. These arrays are equivalent to latin rectangles whose
columns are the blocks of a symmetric balanced design. That this representation is
always possible is a consequence of Philip Hall’s 1935 theorem on systems of distinct
representatives (or, equivalently, König’s theorem), widely known as the Marriage
Problem, which has had many applications in the study of designs.
In 1938, Bose [300] proved that the existence of a complete set of n − 1 MOLS of
order n is equivalent to the existence of a finite projective plane of order n. Then in
1938, Bose and Nair [306] introduced the ideas of association schemes and partially
balanced incomplete block designs. In 1942, Bose [303] gave nontrivial examples of
balanced incomplete block designs with repeated blocks. The term association scheme
was introduced by Bose and Shimamoto [307] in 1952. Bose moved to the US in 1949,
and later with Shrikhande and Parker established the falsity of the Euler conjecture,
as mentioned above. This work showed clearly the usefulness of pairwise balanced
designs.
While Bose’s 1939 paper exploits the algebra of finite fields, the connection with
algebra explored at the time was much broader. In 1877, Cayley [444, 445] discussed
the structure of the multiplication table of a group (its Cayley table). Schröder, from
1890 through 1905, published a two-thousand page treatise [1857] on algebras with
a binary operation, and extensively discussed quasigroups with various restrictions.
Frolov [834] and Schönhardt [1853] exploited the connection between latin squares and
quasigroups. However, only in the 1930s and 1940s did the area take form. In 1935,
Moufang [1640] studied a specific class of quasigroups, introducing in the process the
Moufang loops. In 1937, Ore and Hausmann [1701] first used the term quasigroup in](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-48-320.jpg)
![THE RELATIONS BETWEEN ITALY AND THE
UNITED STATES
Speech by the American Ambassador to Rome.
On the 28th June 1923 the Italo-American Association held in Rome a banquet
in honour of Mr. Richard Washburn Child, American Ambassador to Italy, and of
the Hon. Mussolini, President of the Italian Council. The two distinguished guests
delivered the following speeches,[14] which have a special importance, both with
regard to Fascismo and to Italo-American relations.
14. The two speeches have been courteously given at his request to Baron
Quaranta di San Severino for publication by the American Ambassador, Richard
Washburn Child.
The object of this meeting was clearly explained by the Hon. Baron Sardi, Italian
Under-Secretary of State for Public Works, in an appropriate address to the
illustrious guests (published in full by the Bulletin of the Library for American
Studies in Italy, No. 5), in which, after having thanked them in the name of
Senator Ruffini, President of the Association, still detained on account of important
duties in Geneva, and also in the name of the other members, for the honour they
conferred on the Society by their presence, went on to lay stress on the purpose
for which the Association exists, namely, to promote a better reciprocal
understanding between the American and Italian peoples through the manifold
activities of their respective countries.
The Hon. Sardi announced that during the summer months of this year courses
of preparation will be inaugurated again for American students who wish to come
and visit our country and study our language, literature and history, while for next
October, under the patronage of the American Ambassador and the Italian
Premier, with the co-operation of American and Italian professors, special
industrial and commercial courses are in preparation. The American students will
be able to benefit by the use of the valuable library of the Association, which is
daily enriched by the competent work of Commendatore Harry Nelson Gay and his
collaborators.
The Hon. Sardi, after referring to the fraternity of arms, which during the Great
War brought together the soldiers of Italy and America, said that, having returned
now to the peaceful spheres of industry and culture, these forms of effort](https://image.slidesharecdn.com/478-250509091039-9d6061d3/85/Handbook-of-combinatorial-designs-2nd-ed-Edition-Charles-J-Colbourn-74-320.jpg)
Handbook of combinatorial designs 2nd ed Edition Charles J. Colbourn
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- 5. Series Editor KENNETH H. ROSEN DISCRETE MATHEMATICS AND ITS APPLICATIONS Handbookof CombinatorialDesigns SecondEdition C5068_C000.indd 1 09/21/2006 12:48:52 PM
- 6. C5068_C000.indd 2 09/21/2006 12:48:53 PM
- 7. Juergen Bierbrauer, Introduction to Coding Theory Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Leslie Hogben, Handbook of Linear Algebra Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB®, Second Edition Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Series Editor Kenneth H. Rosen, Ph.D. and DISCRETE MATHEMATICS ITS APPLICATIONS C5068_C000.indd 3 09/21/2006 12:48:53 PM
- 8. Continued Titles Charles C. Lindner and Christopher A. Rodgers, Design Theory Hang T. Lau, A Java Library of Graph Algorithms and Optimization Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications Richard A. Mollin, An Introduction to Cryptography, Second Edition Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers Dingyi Pei, Authentication Codes and Combinatorial Designs Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography C5068_C000.indd 4 09/21/2006 12:48:53 PM
- 9. Series Editor KENNETH H. ROSEN DISCRETE MATHEMATICS AND ITS APPLICATIONS Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Handbookof CombinatorialDesigns SecondEdition Edited by Charles J. Colbourn Jeffrey H. Dinitz C5068_C000.indd 5 09/21/2006 12:48:53 PM
- 10. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑10: 1‑58488‑506‑8 (Hardcover) International Standard Book Number‑13: 978‑1‑58488‑506‑1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any informa‑ tion storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For orga‑ nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Handbook of combinatorial designs / edited by Charles J. Colbourn, Jeffrey H. Dinitz. ‑‑ 2nd ed. p. cm. ‑‑ (Discrete mathematics and its applications) ISBN 1‑58488‑506‑8 1. Combinatorial designs and configurations. I. Colbourn, Charles J. II. Dinitz, Jeffrey H., 1952‑ III. Title. IV. Series. QA166.25.H36 2007 511’.6‑‑dc22 2006050484 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C5068_C000.indd 6 09/21/2006 12:48:53 PM
- 11. Preface The Purpose of this Book The CRC Handbook of Combinatorial Designs (Second Edition) is a reference book of combinatorial designs, including constructions of designs, existence results, proper- ties of designs, and many applications of designs. The handbook is not a textbook in- troduction to combinatorial design theory, but rather a comprehensive, easy-to-access reference for facts about designs. Organization of the Handbook The presentation is organized into seven main parts, each divided into chapters. The first part contains a brief introduction and history. The next four parts center around four main classes of combinatorial designs: balanced incomplete block de- signs (II), orthogonal arrays and latin squares (III), pairwise balanced designs (IV), and Hadamard and orthogonal designs (V). Part VI then contains sixty-five shorter chapters (placed in alphabetical order) devoted to other classes of designs—naturally, many of these classes are closely connected with designs in Parts II–V. Part VII gives mathematical and computational background of particular use in design theory. How to Use this Book This handbook is not meant to be used in a sequential way, but rather for direct access to information about relevant topics. Each chapter makes references to other chapters for related information; in particular, the See Also listing at the end of each chapter provides useful links. An extensive index and the table of contents provide first entry points for most material. For readers new to the area, the first chapter, Opening the Door, can assist in finding a suitable starting point. This handbook can be used on its own as a reference book or in conjunction with numerous excellent available texts on the subject. Although material at the research frontier is often included, the handbook emphasizes basic, useful information. Combinatorial design theory has extensive interactions with numerous areas of mathematics: group theory, graph theory, number theory and finite fields, finite ge- ometry, and linear algebra. It also has applications ranging from routine to complex in numerous disciplines: experimental design, coding theory, cryptography, and com- puter science, among others. The handbook treats all of these, providing information for design theorists to understand the applications, and for users of design theory to obtain pointers in to the rest of the handbook, and hence to the mathematics of combinatorial designs. Notes for the User A home page for the CRC Handbook of Combinatorial Designs is being maintained. It is located at http://www.cems.uvm.edu/˜dinitz/hcd.html. Notes, corrections, and updates will be disseminated at that site. Fixed numbering conventions are used. This book has seven main parts numbered using roman numerals I–VII. Each part is divided into chapters, which in turn have sections; all are numbered. So, for example, the third section in the fourth chapter of the second part is §II.4.3. This is the manner to which it is referred when referenced from any other chapter. When referenced from within Chapter II.4, it is referenced
- 12. Preface simply as §4.3. Occasionally, sections contain subsections. However, to avoid four level numbers, such subsections are not numbered; rather, the beginning of a subsec- tion is given by the symbol ‘B’ preceding the title of the subsection. Numbering of Theorems, Definitions, Remarks, Corollaries, Tables, Algorithms, and the like is con- secutive within each chapter. So, for example, in §II.4.2, Theorem II.4.8 (displayed with “4.8” exdented) is followed by Definition II.4.9, which is followed by Remark II.4.10. As before, the roman numeral is omitted if both the referenced theorem and the reference to it are in the same chapter. Definitions are displayed in gray boxes. Forward and reverse citations are given. There is a single bibliography after Part VII. However, each chapter provides a list of the reference numbers for items cited therein. In addition, the last entry of each item in the bibliography (in angle brackets) enumerates the page numbers on which the item is cited. This is often useful to uncover further relationships among the topics treated. Original references are not, in general, given. Instead, each chapter references surveys and textbooks in which the information is readily accessible. These sources should be consulted for more complete lists of references. One exception is Chapter I.2, Design Theory: Antiquity to 1950, which provides original references. Each chapter covers the basic material, but sufficient pointers to find more specialized information are typically given. The state of the subject is continually advancing. Each chapter presents, as far as possible, information that is current at the time of writing. Inevitably, improvements will occur! Readers aware of improvements on, or corrections to, the material pre- sented here are invited to contact the author(s) of the chapter(s) involved and also the editors-in-chief. Differences Between the Editions The second edition exhibits a substantial increase in size over the first; the book is over 30% larger, with 235 additional pages! While the basic structure has been retained, a new introductory part has been added to provide a general overview and a historical perspective. Material on Hadamard and related designs has been collected together into a new part (V). On the other hand, the Applications part of the first edition has been incorporated into the remaining parts, in the belief that the effort to distinguish theory from application is no longer appropriate. Finally, the nearly 90 separate bibliographies of the first edition have been amalgamated to form a single bibliography for the Handbook, to provide yet another means to find the right “starting point” on a search for desired information. This bibliography contains roughly 2200 references and in itself provides an excellent resource. Often material that is closely related has been combined into a single chapter; in the process, virtually all of the content of the first edition is preserved, but the reader may find small changes in its location. The material is completely updated, and generally enlarged, in the second. The major tables from the first edition (BIBDs, MOLS, PBDs, Hadamard matrices) have been definitive sources for the current knowledge, and all are updated in the second edition. We do not believe that designs whose heyday seems to have passed deserve less attention than the topics of most active current research. The basics of yesteryear continue to be important. Nevertheless, because the discipline continues to evolve, many new topics have been incorporated. Particular emphasis is on topics that bridge with many areas of application, as well as with other areas of mathematics. In the second edition, the new chapters are: Opening the Door (I.1), Design The- ory: Antiquity to 1950 (I.2), Triple Systems (II.2), Group Divisible Designs (IV.4), Optical Orthogonal Codes (V.9), Bent Functions (VI.4), Block-Transitive Designs (VI.5), Covering Arrays (VI.10), Deletion-Correcting Codes (VI.14), Graph Embed-
- 13. Preface dings and Designs (VI.25), Grooming (VI.27), Infinite Designs (VI.30), Low Density Parity Check Codes (VI.33), Magic Squares (VI.34), Nested Designs (VI.36), Perfect Hash Families (VI.43), Permutation Codes and Arrays (VI.44), Permutation Polyno- mials (VI.45), Pooling Designs (VI.46), Quasi-3 Designs (VI.47), Supersimple Designs (VI.57), Turán Systems (VI.61), Divisible Semiplanes (VII.3), Linear Algebra and De- signs (VII.7), Designs and Matroids (VII.10), and Directed Strongly Regular Graphs (VII.12). We are pleased to have been able to add much new material to the Handbook. A Note of Appreciation We must begin by acknowledging the excellent work done by the 110 contributors. Each kindly agreed to write a chapter on a specific topic, their job made difficult by the style and length restrictions placed upon them. Many had written hundreds of pages on their topic, yet were asked to summarize in a few pages. They also were asked to do this within the fairly rigid stylistic framework that is relatively consistent throughout the book. The editors were merciless in their attempt to uphold this style, and offer a special thanks to the copyeditor, Carole Gustafson, for assistance. Many chapters were edited extensively. We thank the contributors for their cooperation. Many chapters have profited from comments from a wide variety of readers, in addition to the contributors and editors. A compilation of this size can only be accomplished with the help of a small army of people. The concept developed and realized in the first edition from 1996 was the cumulative effort of approximately one hundred people. Many have extensively updated their earlier chapters, and some generously agreed to contribute new material. Many chapters in the second edition employ a conceptual design and/or material first written by others. We thank again authors from both editions for worrying less about whose name is on the byline for the material than the fact that it is a good reflection on our discipline. The concept of the Handbook owes much to those who compiled extensive tables prior to the first edition, such as the BIBD tables of Rudi Mathon and Alex Rosa, and the MOLS table of Andries Brouwer. For the first edition, Ron Mullin coordinated a substantial effort to make the first tables of PBDs, and Don Kreher generated extensive tables of t-designs; in the process, both developed new and useful ways to present these tables, updated in the second edition. We thank Ron and Don again for their great contributions. As in the first edition, we again thank Debbie Street and Vladimir Tonchev for their extensive help in organizing the material in statistical designs and coding the- ory, respectively. For the second edition, we especially thank Hadi Kharaghani for organizing (and writing much of) the material in Part V. Special thanks too to Leo Storme for the complete rewrite and reorganization of the chapter on finite geometry. For going above and beyond with their help on the second edition, we also thank Julian Abel, Dan Archdeacon, Frank Bennett, Ian Blake, Warwick de Launey, Peter Dukes, Gennian Ge, Malcolm Greig, Jonathan Jedwab, Teresa Jin, Esther Lamken, Bill Martin, Gary McGuire, Patric Östergård, David Pike, Donald Preece, Alex Rosa, Joe Rushanan, Doug Stinson, Anne Street, Tran van Trung, Ian Wanless, and Zhu Lie. Thanks also to the excellent logistical support from the publisher, particularly to Nora Konopka and Wayne Yuhasz for the first edition, and to Helena Redshaw, Judith Simon, and Bob Stern for the second. Finally, we thank Sue Dinitz and Violet Syrotiuk for their love and support. Putting together this extensive volume has taken countless hours; we thank them for their patience during it all. Charles J. Colbourn Jeffrey H. Dinitz
- 15. To Violet, Sarah, and Susie, with love. CJC To my father, Simon Dinitz, who has inspired me in all that I have done. Happy 80th birthday! JHD
- 17. Editors-in-Chief Charles J. Colbourn Computer Science and Engineering Arizona State University U.S.A. Jeffrey H. Dinitz Mathematics and Statistics University of Vermont U.S.A Contributors R. Julian R. Abel University of New South Wales Australia Lars D. Andersen University of Aalborg Denmark Ian Anderson University of Glasgow U.K. Lynn Margaret Batten Deakin University Australia Lucien Bénéteau INSA France Frank E. Bennett Mount Saint Vincent University Canada Jean-Claude Bermond INRIA Sophia-Antipolis France Anton Betten Colorado State University U.S.A. Jürgen Bierbrauer Michigan Technological University U.S.A. Ian F. Blake University of Toronto Canada Andries E. Brouwer Technical University of Eindhoven Netherlands Darryn Bryant University of Queensland Australia Marco Buratti Università di Perugia Italy Peter J. Cameron University of London U.K.
- 18. Contributors Yeow Meng Chee Card View Pte Ltd. Singapore Leo G. Chouinard II University of Nebraska U.S.A. Wensong Chu Arizona State University U.S.A. Charles J. Colbourn Arizona State University U.S.A. David Coudert INRIA Sophia-Antipolis France Robert Craigen University of Manitoba Canada Anne Delandtsheer Université Libre de Bruxelles Belgium Warwick de Launey Center for Communications Research U.S.A. Michel M. Deza École Normale Supérieure France Jeffrey H. Dinitz University of Vermont U.S.A. Peter J. Dukes University of Victoria Canada Saad El-Zanati Illinois State University U.S.A. Anthony B. Evans Wright State University U.S.A. Norman J. Finizio University of Rhode Island U.S.A. Dalibor Froncek University of Minnesota Duluth U.S.A. Gennian Ge Zhejiang University P.R. China Peter B. Gibbons University of Auckland New Zealand Chistopher D. Godsil University of Waterloo Canada Solomon W. Golomb University of Southern California U.S.A. K. Gopalakrishnan East Carolina University U.S.A. Daniel M. Gordon Center for Communications Research U.S.A. Ken Gray The University of Queensland Australia Malcolm Greig Greig Consulting Canada Terry S. Griggs The Open University U.K. Hans-Dietrich O. F. Gronau Universität Rostock Germany Harald Gropp Universität Heidelberg Germany
- 19. Contributors A. S. Hedayat University of Illinois at Chicago U.S.A. Tor Helleseth University of Bergen Norway Sylvia A. Hobart University of Wyoming U.S.A. Spencer P. Hurd The Citadel U.S.A. Frank K. Hwang National Chiao Tung University Taiwan Yury J. Ionin Central Michigan University U.S.A. Robert Jajcay Indiana State University U.S.A. Dieter Jungnickel Universität Augsburg Germany Hadi Kharaghani University of Lethbridge Canada Gholamreza B. Khosrovshahi IPM Iran Christos Koukouvinos National Technical University of Athens Greece Earl S. Kramer University of Nebraska–Lincoln U.S.A. Donald L. Kreher Michigan Technological University U.S.A. Joseph M. Kudrle University of Vermont U.S.A. Esther R. Lamken San Francisco U.S.A. Reinhard Laue Universität Bayreuth Germany Charles F. Laywine Brock University Canada Vladimir I. Levenshtein Keldysh Inst. Applied Mathematics Russia P. C. Li University of Manitoba Canada Charles C. Lindner Auburn University U.S.A. Spyros S. Magliveras Florida Atlantic University U.S.A. Alireza Mahmoodi York University Canada William J. Martin Worcester Polytechnic Institute U.S.A. Rudolf Mathon University of Toronto Canada Gary McGuire National University of Ireland Ireland Sarah B. Menard University of Vermont U.S.A.
- 20. Contributors Eric Mendelsohn University of Toronto Canada Ying Miao University of Tsukuba Japan J. P. Morgan Virginia Tech U.S.A. Gary L. Mullen The Pennsylvania State University U.S.A. Ronald C. Mullin Florida Atlantic University U.S.A. Akihiro Munemasa Tohoku University Japan William Orrick Indiana University U.S.A. Patric R. J. Östergård Helsinki University of Technology Finland Stanley E. Payne University of Colorado at Denver U.S.A. Alexander Pott Universität Magdeberg Germany Donald A. Preece University of London U.K. Chris Rodger Auburn University U.S.A. Alexander Rosa McMaster University Canada Gordon F. Royle University of Western Australia Australia Miklós Ruszinkó Hungarian Academy of Sciences Hungary Dinesh G. Sarvate College of Charleston U.S.A. Nabil Shalaby Memorial University of Newfoundland Canada James B. Shearer IBM U.S.A. Mohan S. Shrikhande Central Michigan University U.S.A. Jozef Širáň University of Auckland New Zealand Ken W. Smith Central Michigan University U.S.A. Hong-Yeop Song Yonsei University Korea Sung Y. Song Iowa State University U. S. A. Edward Spence University of Glasgow U.K. Douglas R. Stinson University of Waterloo Canada Leo Storme Ghent University Belgium
- 21. Contributors Anne Penfold Street The University of Queensland Australia Deborah J. Street University of Technology, Sydney Australia Herbert Taylor University of Southern California U.S.A. Joseph A. Thas Ghent University Belgium Vladimir D. Tonchev Michigan Technological University U.S.A. David C. Torney Los Alamos National Laboratory U.S.A. G. H. John van Rees University of Manitoba Canada Tran van Trung University of Duisburg–Essen Germany Robert A. Walker II Arizona State University U.S.A. Walter D. Wallis Southern Illinois University U.S.A. Ian M. Wanless Monash University Australia Bridget S. Webb The Open University U.K. Ruizhong Wei Lakehead University Canada Hugh Williams University of Calgary Canada Richard M. Wilson California Institute of Technology U.S.A. Jianxing Yin Suzhou University China L. Zhu Suzhou University China
- 23. Contents I Introduction 1 Opening the Door Charles J. Colbourn 3 2 Design Theory: Antiquity to 1950 Ian Anderson, Charles J. Colbourn, Jeffrey H. Dinitz, Terry S. Griggs 11 II Block Designs 1 2-(v, k, λ) Designs of Small Order Rudolf Mathon, Alexander Rosa 25 2 Triple Systems Charles J. Colbourn 58 3 BIBDs with Small Block Size R. Julian R. Abel, Malcolm Greig 72 4 t-Designs with t ≥ 3 Gholamreza B. Khosrovshahi, Reinhard Laue 79 5 Steiner Systems Charles J. Colbourn, Rudolf Mathon 102 6 Symmetric Designs Yury J. Ionin, Tran van Trung 110 7 Resolvable and Near-Resolvable Designs R. Julian R. Abel, Gennian Ge, Jianxing Yin 124 III Latin Squares 1 Latin Squares Charles J. Colbourn, Jeffrey H. Dinitz, Ian M. Wanless 135 2 Quasigroups Frank E. Bennett, Charles C. Lindner 152
- 24. Contents 3 Mutually Orthogonal Latin Squares (MOLS) R. Julian R. Abel, Charles J. Colbourn, Jeffrey H. Dinitz 160 4 Incomplete MOLS R. Julian R. Abel, Charles J. Colbourn, Jeffrey H. Dinitz 193 5 Self-Orthogonal Latin Squares (SOLS) Norman J. Finizio, L. Zhu 211 6 Orthogonal Arrays of Index More Than One Malcolm Greig, Charles J. Colbourn 219 7 Orthogonal Arrays of Strength More Than Two Charles J. Colbourn 224 IV Pairwise Balanced Designs 1 PBDs and GDDs: The Basics Ronald C. Mullin, Hans-Dietrich O. F. Gronau 231 2 PBDs: Recursive Constructions Malcolm Greig, Ronald C. Mullin 236 3 PBD-Closure R. Julian R. Abel, Frank E. Bennett, Malcolm Greig 247 4 Group Divisible Designs Gennian Ge 255 5 PBDs, Frames, and Resolvability Gennian Ge, Ying Miao 261 6 Pairwise Balanced Designs as Linear Spaces Anton Betten 266 V Hadamard Matrices and Related Designs 1 Hadamard Matrices and Hadamard Designs Robert Craigen, Hadi Kharaghani 273 2 Orthogonal Designs Robert Craigen, Hadi Kharaghani 280 3 D-Optimal Matrices Hadi Kharaghani, William Orrick 296 4 Bhaskar Rao Designs Warwick de Launey 299 5 Generalized Hadamard Matrices Warwick de Launey 301
- 25. Contents 6 Balanced Generalized Weighing Matrices and Conference Matrices Yury J. Ionin, Hadi Kharaghani 306 7 Sequence Correlation Tor Helleseth 313 8 Complementary, Base, and Turyn Sequences Hadi Kharaghani, Christos Koukouvinos 317 9 Optical Orthogonal Codes Tor Helleseth 321 VI Other Combinatorial Designs 1 Association Schemes Christopher D. Godsil, Sung Y. Song 325 2 Balanced Ternary Designs Spencer P. Hurd, Dinesh G. Sarvate 330 3 Balanced Tournament Designs Esther R. Lamken 333 4 Bent Functions Douglas R. Stinson 337 5 Block-Transitive Designs Anne Delandtsheer 339 6 Complete Mappings and Sequencings of Finite Groups Anthony B. Evans 345 7 Configurations Harald Gropp 353 8 Correlation-immune and Resilient Functions K. Gopalakrishnan, Douglas R. Stinson 355 9 Costas Arrays Herbert Taylor, Jeffrey H. Dinitz 357 10 Covering Arrays Charles J. Colbourn 361 11 Coverings Daniel M. Gordon, Douglas R. Stinson 365 12 Cycle Decompositions Darryn Bryant, Chris Rodger 373 13 Defining Sets Ken Gray, Anne Penfold Street 382
- 26. Contents 14 Deletion-correcting Codes Vladimir I. Levenshtein 385 15 Derandomization K. Gopalakrishnan, Douglas R. Stinson 389 16 Difference Families R. Julian R. Abel, Marco Buratti 392 17 Difference Matrices Charles J. Colbourn 411 18 Difference Sets Dieter Jungnickel, Alexander Pott, Ken W. Smith 419 19 Difference Triangle Sets James B. Shearer 436 20 Directed Designs Frank E. Bennett, Alireza Mahmoodi 441 21 Factorial Designs Deborah J. Street 445 22 Frequency Squares and Hypercubes Charles F. Laywine, Gary L. Mullen 465 23 Generalized Quadrangles Stanley E. Payne 472 24 Graph Decompositions Darryn Bryant, Saad El-Zanati 477 25 Graph Embeddings and Designs Jozef Širáň 486 26 Graphical Designs Yeow Meng Chee, Donald L. Kreher 490 27 Grooming Jean-Claude Bermond, David Coudert 494 28 Hall Triple Systems Lucien Bénéteau 496 29 Howell Designs Jeffrey H. Dinitz 499 30 Infinite Designs Peter J. Cameron, Bridget S. Webb 504 31 Linear Spaces: Geometric Aspects Lynn Margaret Batten 506
- 27. Contents 32 Lotto Designs (Ben) P. C. Li, G. H. John van Rees 512 33 Low Density Parity Check Codes Ian F. Blake 519 34 Magic Squares Joseph M. Kudrle, Sarah B. Menard 524 35 Mendelsohn Designs Eric Mendelsohn 528 36 Nested Designs J. P. Morgan 535 37 Optimality and Efficiency: Comparing Block Designs Deborah J. Street 540 38 Ordered Designs, Perpendicular Arrays, and Permutation Sets Jürgen Bierbrauer 543 39 Orthogonal Main Effect Plans Deborah J. Street 547 40 Packings Douglas R. Stinson, Ruizhong Wei, Jianxing Yin 550 41 Partial Geometries Joseph A. Thas 557 42 Partially Balanced Incomplete Block Designs Deborah J. Street, Anne Penfold Street 562 43 Perfect Hash Families Robert A. Walker II, Charles J. Colbourn 566 44 Permutation Codes and Arrays Peter J. Dukes 568 45 Permutation Polynomials Gary L. Mullen 572 46 Pooling Designs David C. Torney 574 47 Quasi-3 Designs Gary McGuire 576 48 Quasi-Symmetric Designs Mohan S. Shrikhande 578 49 (r, λ)-designs G. H. John van Rees 582
- 28. Contents 50 Room Squares Jeffrey H. Dinitz 584 51 Scheduling a Tournament J. H. Dinitz, Dalibor Froncek, Esther R. Lamken, Walter D. Wallis 591 52 Secrecy and Authentication Codes K. Gopalakrishnan, Douglas R. Stinson 606 53 Skolem and Langford Sequences Nabil Shalaby 612 54 Spherical Designs Akihiro Munemasa 617 55 Starters Jeffrey H. Dinitz 622 56 Superimposed Codes and Combinatorial Group Testing Charles J. Colbourn, Frank K. Hwang 629 57 Supersimple Designs Hans-Dietrich O. F. Gronau 633 58 Threshold and Ramp Schemes K. Gopalakrishnan, Douglas R. Stinson 635 59 (t,m,s)-Nets William J. Martin 639 60 Trades A. S. Hedayat, Gholamreza B. Khosrovshahi 644 61 Turán Systems Miklós Ruszinkó 649 62 Tuscan Squares Wensong Chu, Solomon W. Golomb, Hong-Yeop Song 652 63 t-Wise Balanced Designs Earl S. Kramer, Donald L. Kreher 657 64 Whist Tournaments Ian Anderson, Norman J. Finizio 663 65 Youden Squares and Generalized Youden Designs Donald A. Preece, Charles J. Colbourn 668 VII Related Mathematics 1 Codes Vladimir D. Tonchev 677
- 29. Contents 2 Finite Geometry Leo Storme 702 3 Divisible Semiplanes Rudolf Mathon 729 4 Graphs and Multigraphs Gordon F. Royle 731 5 Factorizations of Graphs Lars D. Andersen 740 6 Computational Methods in Design Theory Peter B. Gibbons, Patric R. J. Östergård 755 7 Linear Algebra and Designs Peter J. Dukes, Richard M. Wilson 783 8 Number Theory and Finite Fields Jeffrey H. Dinitz, Hugh C. Williams 791 9 Finite Groups and Designs Leo G. Chouinard II, Robert Jajcay, Spyros S. Magliveras 819 10 Designs and Matroids Peter J. Cameron, Michel M. Deza 847 11 Strongly Regular Graphs Andries E. Brouwer 852 12 Directed Strongly Regular Graphs Andries E. Brouwer, Sylvia A. Hobart 868 13 Two-Graphs Edward Spence 875 Bibliography and Index Bibliography 883 Index 967
- 33. I.1 Opening the Door 3 It seems to me that whatever else is beautiful apart from absolute beauty is beautiful because it partakes of that absolute beauty, and for no other reason. Do you accept this kind of causality? Yes, I do. Well, now, that is as far as my mind goes; I cannot understand these other ingenious theories of causation. If someone tells me that the reason why a given object is beautiful is that it has a gorgeous color or shape or any other such attribute, I disregard all these other explanations—I find them all confusing—and I cling simply and straightforwardly and no doubt foolishly to the explanation that the one thing that makes the object beautiful is the presence in it or association with it, in whatever way the relation comes about, of absolute beauty. I do not go so far as to insist upon the precise details—only upon the fact that it is by beauty that beautiful things are beautiful. This, I feel, is the safest answer for me or anyone else to give, and I believe that while I hold fast to this I cannot fall; it is safe for me or for anyone else to answer that it is by beauty that beautiful things are beautiful. Don’t you agree? (Plato, Phaedo, 100c-e) 1 Opening the Door Charles J. Colbourn 1.1 Example The Fano plane. Seven points Three points on a line Every two points define a line Seven lines Three lines through a point Every two lines meet at a point 1.2 Remarks Opening the door into the world of combinatorial designs, and stepping through that door, is familiar to some and mysterious to many. What lies on the other side? And where is the map to guide us? There is territory both charted and uncharted; the guide map is at best known only in part. Meandering through this world without fixed direction can be both enjoyable and instructive; yet most have a goal, a question whose answer is sought. This introduction explores, with the simplest examples, the landmarks of the world of combinatorial designs. While not prescribing a path, it provides some direction. Perhaps the most vexing problem is to recognize the object sought when it is encountered; the disguises are often ingenious, but the many faces of one simple object often reveal its importance. Example 1.1 is recast in many different ways, to acquaint the reader with correspondences that are useful in recognizing an object when its representation changes. This exercise provides a first view of the large body of material in the rest of the handbook, to which pointers are provided for the reader to follow. 1.1 Balance B Sets and Blocks 1.3 Example The Fano plane as a set system. {0,1,2}, {0,3,4}, {0,5,6}, {1,3,5}, {1,4,6}, {2,3,6}, {2,4,5} Using elements {0, 1, 2, 3,4,5, 6}, label the points in Example 1.1. For each line, form a set that contains the elements corre- sponding to the points on that line. The result is a collection of sets, a set system.
- 34. 4 Opening the Door I.1 1.4 Remark The set system of Example 1.3 exhibits in many ways a desideratum that is central to the understanding of combinatorial designs, the desire for balance. Some of the ways are considered here. • All sets have the same size k = 3. The set system is k-uniform. • All elements occur in precisely r = 3 of the sets. The set system is r-regular or equireplicate with replication r. • Every pair of distinct elements occurs as a subset of exactly λ = 1 of the sets. The set system is 2-balanced with index λ = 1. • Every pair of distinct sets intersects in exactly µ = 1 elements. Equivalently, the symmetric difference of every two distinct sets contains exactly four elements; the set system is 4-equidistant. Balance is a key to understanding combinatorial designs, not just a serendipitous feature of one small example. 1.5 Consider a general finite set system. It has a (finite) ground set V of size v whose members are elements (or points or varieties), equipped with a (finite) collection B of b subsets of V whose members are blocks (or lines or treatments). Then • K = {|B| : B ∈ B} is the set of block sizes; • R = {|{B : x ∈ B ∈ B}| : x ∈ V } is the set of replication numbers, • for t ≥ 0 integer, Λt = {|{B : T ⊆ B ∈ B}| : T ⊆ V, |T| = t} is the set of t-indices. Indeed, Λ0 = {b}, and Λ1 = R. • M = {|B ∩ B0 | : B, B0 ∈ B, B 6= B0 }, the set of intersection numbers. 1.6 Remark Restricting any or all of K, R, M, or the {Λt}s imposes structure on the set system; restrictions to a single element balance that structure in some regard. Table 1.7 considers the effect of restricting one or more of these sets to be singletons, providing pointers into the remainder of the handbook. Indeed, set systems so defined are among the most central objects in all of design theory and are studied extensively. The language with which they are discussed in this chapter varies from set systems to hypergraphs, matrices, geometries of points and lines, graph decompositions, codes, and the like; the balance of the underlying set system permeates all. 1.7 Table Some of the set systems that result when at least one of the balance sets K, R, M, or the {Λt} is a singleton. K R Λ2 Λt M Other Name Section {k} {r} {λ2} {λt} t-design II.4 {k} {r} {λ2} {λt} λt = 1 Steiner system II.5 {k} {r} {λ} balanced incomplete block de- sign (BIBD) II.1, II.3 {k} {r} {λ} {µ} k = r symmetric design (SBIBD) II.6 {k} {r} {λ} {µ} λ = µ = 1 projective plane II.6, VII.2.1 {r} {λ} (r, λ)-design VI.49 {λ} pairwise balanced design (PBD) IV.1 {λ} λ = 1 linear space IV.1, IV.6 {λt} t-wise balanced design VI.63 1.8 Remarks A natural relaxation is obtained by specifying an upper or a lower bound on values arising in the set, or by specifying a small number of permitted values; the short form {x} is adopted to permit all values greater than or equal to x, and {x} to indicate all values less than or equal to x. Then certain generalizations have been well studied; see Table 1.9.
- 35. I.1.1 Balance 5 1.9 Table Some set systems that result when K is a singleton and M or {Λt} is bounded. K R Λ2 Λt M Other Name Section {k} {λt} t-covering VI.11 {k} {µ} constant weight code VII.1 {k} {λt} t-packing VI.40 B Algebras and Arrays 1.10 Example A quasigroup and a latin square. ⊗ 0 1 2 3 4 5 6 0 0 2 1 4 3 6 5 1 2 1 0 5 6 3 4 2 1 0 2 6 5 4 3 3 4 5 6 3 0 1 2 4 3 6 5 0 4 2 1 5 6 3 4 1 2 5 0 6 5 4 3 2 1 0 6 Use the blocks of Example 1.3 to define the binary operation ⊗ at left, as follows. When {x, y, z} is a block, define x ⊗ y = z; then define x ⊗ x = x. Delete the headline and sideline, reorder rows and columns, and rename symbols to get the array on the right. 1 7 2 6 5 4 3 4 5 6 3 7 1 2 3 6 5 7 4 2 1 5 4 3 2 1 7 6 2 1 7 5 6 3 4 7 2 1 4 3 6 5 6 3 4 1 2 5 7 1.11 Remarks Example 1.10 introduces another kind of balance. The operation ⊗ defined has a simple property: the equation a⊗b = c always has a unique solution for the third variable given the other two. This is reminiscent of groups, yet the algebra defined has no identity element and hence no notion of inverses. It weakens the properties of finite groups, and is a quasigroup. As an n×n array, it is equivalent to say that every symbol occurs in every row once, and also in every column once; the array is a latin square. Part III focuses on latin squares and their generalizations; see particularly §III.1 for latin squares, §III.2 for quasigroups, and §VI.22 for generalizations to balanced squares in which every cell contains a fixed number of elements greater than one. 1.12 Example For the square at left in Example 1.10, form a set {x, y+7, z+14} whenever x ⊗ y = z. The resulting sets are: {0, 7, 14}, {0, 8, 16}, {0, 9, 15}, {0, 10, 18}, {0, 11, 17}, {0, 12, 20}, {0, 13, 19}, {1, 7, 16}, {1, 8, 15}, {1, 9, 14}, {1, 10, 19}, {1, 11, 20}, {1, 12, 17}, {1, 13, 18}, {2, 7, 15}, {2, 8, 14}, {2, 9, 16}, {2, 10, 20}, {2, 11, 19}, {2, 12, 18}, {2, 13, 17}, {3, 7, 18}, {3, 8, 19}, {3, 9, 20}, {3, 10, 17}, {3, 11, 14}, {3, 12, 15}, {3, 13, 16}, {4, 7, 17}, {4, 8, 20}, {4, 9, 19}, {4, 10, 14}, {4, 11, 18}, {4, 12, 16}, {4, 13, 15}, {5, 7, 20}, {5, 8, 17}, {5, 9, 18}, {5, 10, 15}, {5, 11, 16}, {5, 12, 19}, {5, 13, 14}, {6, 7, 19}, {6, 8, 18}, {6, 9, 17}, {6, 10, 16}, {6, 11, 15}, {6, 12, 14}, {6, 13, 20}. 1.13 Remarks Example 1.12 gives a representation of the quasigroup in Example 1.10 once again as blocks. In this set system with 49 blocks on 21 points, every point occurs in 7 blocks, every block has three points; yet some pairs occur once while others occur not at all, in the blocks. Indeed pairs inside the groups {0, 1, 2, 3,4,5,6}, {7, 8, 9, 10, 11, 12, 13}, and {14, 15, 16, 17,18, 19, 20} are precisely those that do not appear. This is a group divisible design (see §IV.1). Balance with respect to occurrence of pairs could be recovered by adjoining the three groups as blocks; the result exhibits pair balance but loses equality of block sizes because blocks of size 3 and 7 are present; this is a pairwise balanced design (see §IV.1). Instead, placing a copy of the 7 point design on each of the groups balances both pair occurrences and block sizes, and, indeed, provides a larger balanced incomplete block design, this one on 21 points having 70 (= 49 + 3 × 7) blocks. Returning to the group divisible design, yet another balance property is found; every block meets every group in exactly one point. This defines a transversal design; see §III.3.
- 36. 6 Opening the Door I.1 1.14 Example Two mutually orthogonal latin squares of side 7. + 0 1 2 3 4 5 6 0 0 1 2 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 4 5 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 1 2 3 5 5 6 0 1 2 3 4 6 6 0 1 2 3 4 5 0 1 2 3 4 5 6 0 0 2 4 6 1 3 5 1 1 3 5 0 2 4 6 2 2 4 6 1 3 5 0 3 3 5 0 2 4 6 1 4 4 6 1 3 5 0 2 5 5 0 2 4 6 1 3 6 6 1 3 5 0 2 4 +, 0 1 2 3 4 5 6 0 0,0 1,2 2,4 3,6 4,1 5,3 6,5 1 1,1 2,3 3,5 4,0 5,2 6,4 0,6 2 2,2 3,4 4,6 5,1 6,3 0,5 1,0 3 3,3 4,5 5,0 6,2 0,4 1,6 2,1 4 4,4 5,6 6,1 0,3 1,5 2,0 3,2 5 5,5 6,0 0,2 1,4 2,6 3,1 4,3 6 6,6 0,1 1,3 2,5 3,0 4,2 5,4 1.15 Remark The operation ⊗ in Example 1.10 is by no means the only way to define a quasigroup on 7 symbols. In Example 1.14, two more are shown, with operations + and . A natural question is whether they are different from each other in an essential manner, or different from ⊗. The definition of + has the same columns as that of , but in a different order. While different as algebras, they are nonetheless quite similar. They are introduced here to note yet another kind of balance. When superimposing two n × n latin squares, one could expect every ordered pair of symbols to arise any number of times between 0 and n; but the average number of times is precisely one. This example demonstrates that perfect balance can be achieved: every pair arises exactly once in the superposition. As latin squares, these are mutually orthogonal. Section III.3 explores these, in their various disguises as quasigroups, as squares, and as transversal designs. 1.16 Example For the square at left in Example 1.10, form an array whose columns are precisely the triples (x, y, z)T for which x ⊗ y = z. 0000000111111122222223333333444444455555556666666 0123456012345601234560123456012345601234560123456 0214365210563410265434563012365042163412505432106 1.17 Remark Example 1.16 gives yet another view of the same structure. In the 3 × 49 array, each of the 7 symbols occurs 7 times in each row. However, much more is true. Choosing any two rows, there are 49 = 7·7 pairs of entries possible, and 49 occurring. As one might expect, the case of interest is that of balance, when every pair occurs the same number of times (here, once). This is an orthogonal array; see §III.6 and §III.7. While playing a prominent role in combinatorial design theory, they play a larger role yet in experimental design in statistics; see §VI.21. B Matrices, Vectors, and Codes 1.18 Example Begin with the set system representation of the Fano plane in Example 1.3 and construct the 7 × 7 matrix of 0s and 1s on the right by placing a 1 in cell (a, b) if the element a is in set b; place a 0 otherwise. 1.19 Remark The matrix in Example 1.18 is the incidence ma- trix of the Fano plane. Generally, the rows are indexed by the points of the design and the columns by the blocks. Im- portant properties of designs can be determined by matrix properties of these incidence matrices; see §VII.7. 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1.20 Example Apply the permutation (0)(1)(2 5 3)(4)(6) to the symbols in Example 1.3. Form a 7 × 7 matrix with columns indexed by blocks and rows indexed by symbols.
- 37. I.1.1 Balance 7 − − − − − − −− − − − + − + ++ − + − − + − ++ − + + − − + −+ − + + + − − +− − − + + + − −+ − + − + + + −− − − + − + + +− Place a − in the cell of row x and column y when symbol x appears in block y, and a + otherwise. Finally add an initial row and an initial column consisting entirely of − to get the array at the left. Invert + and − to get the array on the right. + + + + + + ++ + + + − + − −− + − + + − + −− + − − + + − +− + − − − + + −+ + + − − − + +− + − + − − − ++ + + − + − − −+ 1.21 Remark Of course, the balance observed previously repeats in the matrix on the left in Example 1.20. In this presentation, however, the balance is seen in a new light. Taking + to be +1 and − to be −1, it is possible for two rows to have inner product between −8 and +8; yet here every pair of distinct rows have inner product exactly equal to 0 (and indeed, so do any two distinct columns). This property over {±1} defines a Hadamard matrix; see §V.1. Such matrices have the remarkable property that they maximize the matrix determinant over all n×n {±1}-matrices; see §V.3. Generalizing to arithmetic over algebras other than ({±1}, ·) yields classes of orthogonal matrices and designs (§V.2). The matrix on the right shares the same properties, being the negative of the one on the left. But a surprise is in store. Juxtapose the two horizontally, and delete the two constant columns to form an 8 × 14 matrix. Every two distinct columns of the result have either 0 or 4 identical entries. For each column form a set that contains the indices of the + entries in that column; this yields 14 blocks each of size 4, on a set of 8 elements. Every element occurs in 7 blocks, every 2-subset of elements in 6 blocks, and every 3-subset of elements in exactly one block. This is a Steiner system and a 3-design (a Hadamard 3-design). 1.22 Example In Example 1.20, delete the constant row and constant column, and treat the remaining 7 rows as {±1}-vectors. These vectors are: [--+-+++] [+--+-++] [++--+-+] [+++--+-] [-+++--+] [+-+++--] [-+-+++-] 1.23 Remark In communicating between a sender and a receiver, synchronization is a ma- jor issue, as demonstrated by the message ...EATEATEATEATEATEATEATEA..., which might say EAT, ATE, or TEA depending on your timing. Knowing the message, how- ever, enables one to determine the timing. Suppose then that a sender repeatedly sent [+-+++--]. If the receiver is “in phase,” it multiplies the expected vector with the received one, and computes 7. If “out of phase,” it instead computes −1 no matter what the nonzero phase shift is. This autocorrelation can be used to synchronize, and the balance plays a practical purpose; see §V.9 and §V.7. 1.24 Example An error-correcting code. Replace + by 1 and − by 0 in the two Hadamard ma- trices of Example 1.20, and treat the 16 rows of the resulting matrices as binary vectors of length 8. [00000000] [00010111] [01001011] [01100101] [01110010] [00111001] [01011100] [00101110] [11111111] [11101000] [10110100] [10011010] [10001101] [11000110] [10100011] [11010001] 1.25 Remark The sixteen binary vectors of Example 1.24 have the property that every two distinct vectors differ in exactly 4 or in all 8 positions. A sender and receiver can agree that in order to communicate sixteen information symbols, they are to use these 16 “codewords.” If a single bit error is made in transmission, there is exactly one codeword that could have been intended; if two bit errors are made, the receiver knows that an error occurred but does not know which two bit errors account for the incorrect received word. This is a 1-error-correcting code and a 2-error-detecting code; see §VII.1.
- 38. 8 Opening the Door I.1 B Graphs 1.26 Example An embedding of the Fano plane on the torus. 0 0 0 0 1 2 4 6 3 5 Draw the complete graph on 7 vertices (K7) on the torus (identify the left and right borders, and the top and bot- tom borders, to form the torus—you may need to pur- chase a second Handbook to do this). The number of vertices is 7 and the number of edges is 21, so the num- ber of faces (by Euler’s formula) is 14. 1.27 Remark The triangular faces in the drawing of Example 1.26 have the property that they can be colored with two colors, so that every edge belongs to one face of each color. Consider then the faces of one color; every edge (2-subset of vertices) appears in exactly one of the chosen faces (3-subset of vertices). Seven vertices, seven faces, three vertices on every face, three faces at every vertex, and for each edge exactly one face that contains it. See §VI.25 for embeddings of graphs in surfaces that correspond to designs. 1.28 Example In the graph shown, label points on the left by seven different symbols, and each point on the right by the 3-subset of symbols for its adjacent vertices. The seven 3-subsets thereby cho- sen form the blocks of a balanced incomplete block design, and this incidence graph shows the element-block incidence structure. 1.29 Remark Example 1.28 gives a purely graph-theoretic representation of the running example; see §VII.4. Again a surprise is in store. Consider Example 1.26 further. Surface duality implies that, by interchanging the roles of faces and vertices in the drawing, the embedding of a dual graph is obtained. What is the structure of the dual graph in this case? It is shown in Example 1.28. 1.30 Example The graph in Example 1.28 has a hamiltonian cycle. Re- draw the graph with the hamiltonian cycle on the exterior. This is the Heawood graph, the smallest 3-regular graph whose shortest cycle has length six (also called a cage); see §VII.4. 1.31 Example A decomposition of the complete graph into triangles. 1.32 Remark In another vernacular, Example 1.26 shows that K7 can be decomposed into edge-disjoint copies of K3 (the triangle). The toroidal embedding is not needed to see this, however. Example 1.31 explicitly shows seven K3s that edge-partition K7. Another line of investigation opens: When can one edge-partition a graph G into copies of a graph H? Combinatorial design theory concentrates on cases when G is a complete graph. When H is also complete, another view of balanced incomplete block designs is obtained. But when H is a cycle, a different generalization is encountered, cycle systems; see §VI.12. Further, when H is arbitrary, a graph design is obtained; see §VI.24. Imagine that the set of all possible edges is to be partitioned into subgraphs, so that no subgraph has too many edges; in addition, every vertex in a subgraph that has one or more edges at it incurs a cost. The goal is to minimize cost. This is achieved by choosing subgraphs that are complete whenever possible, and hence block designs provide solutions. Exactly such a problem arises in “grooming” traffic in optical networks; see §VI.27.
- 39. I.1.1 Balance 9 1.33 Example Start with blocks {0, 1, 3}, {0, 4, 5}, {0, 2, 6}, {1, 2, 4}, {1, 5, 6}, {2, 3, 5}, and {3, 4, 6}. Arrange the points 0, . . ., 6 on the perimeter of a circle consecutively, and place an additional point, ∞, at the center of the circle. Now form seven graphs numbered also 0, . . ., 6. In the ith, place the edge {∞, i}, and for all x ∈ {0, . . ., 6}{i} place the edge {x, y} if and only if {i, x, y} is a block. 1.34 Remark In Example 1.33, each of the seven graphs is a perfect matching or 1-factor on the eight points. The seven 1-factors share no edges; they partition the edges of K8 into 1-factors. This is a 1-factorization; see §VII.5. Playing a round-robin tournament with 8 players over 7 time-periods can be accomplished using the matches shown here; see §VII.51. 1.35 Example An orthogonal 1-factorization and a Room square. ∞0 26 45 13 ∞1 03 56 24 ∞2 14 06 35 46 ∞3 25 01 05 ∞4 36 12 23 16 ∞5 04 15 34 02 ∞6 Above is shown another 1-factorization, obtained from Example 1.33 by reflecting the first picture through the vertical axis. Remarkably it has the property that every 1-factor of this factorization shares at most one edge with that of Example 1.33, such one factorizations are orthogonal. Form a 7 × 7 square (at left) with rows indexed by the factors of Example 1.33 and columns by the factors in its re- flection. In each cell, place the edge (if any) that the factors share. 1.36 Remark Example 1.35 gives a square in which every cell is either empty or contains an unordered pair (edge); every unordered pair occurs in exactly one cell; and every row and every column contains every symbol exactly once. This is a Room square; see §VI.50. Generalizing to factorizations of regular graphs other than complete ones leads to Howell designs; see §VI.29. B Finite Geometry 1.37 Example In the picture at right, label the three corner points by binary vectors (0,0,1), (0,1,0), and (1,0,0). When (a, b, c) and (x, y, z) are points, so also is (a ⊕ x, b ⊕ y, c ⊕ z) = (a, b, c) ⊕ (x, y, z); the operation ⊕ is addition modulo 2. Then (1,1,0), (1,0,1), (0,1,1), and (1,1,1) are also points—but (0,0,0) is not. Lines are defined by all sets of three distinct points whose sum under ⊕ is (0,0,0). 1.38 Remark This is a geometry of points and lines that is finite; as with plane geometry, any two points define a line, and two lines meet in at most one point. Unlike usual geometry in the plane, however, there are no two lines parallel; every two lines meet in exactly one point. This is a finite projective geometry, one of a number of finite geometries (§VII.2). Geometries in which lines can have different sizes generalize this notion to linear spaces; see §VI.31.
- 40. 10 Opening the Door I.1 1.2 Symmetry 1.39 Remark That the one simple structure in Example 1.1 can demonstrate balance in so many different settings is remarkable, and its vast generalizations through these different representations underpin many topics in combinatorial design theory. But the example leads to much more. It exhibits symmetry as well as balance. As a first step, note that permuting the three coordinate positions in Example 1.37 relabels the points. Yet the picture remains unchanged! This describes a sixfold symmetry. 1.40 Example In Z7 form the 3-set {1, 2, 4} consisting of the three nonzero squares (the quadratic residues). Form seven sets by adding each element of Z7 in turn to this base block. These seven sets give the set system representation of the Fano plane. 1.41 Remark Example 1.40 gives another representation of the recurring example, as trans- lates of the quadratic residues modulo 7; this opens the door into number theory (§VII.8). More than that, there is a sevenfold symmetry by translation through Z7. Example 1.31 depicted this symmetry already, but in Example 1.40 a more compact representation is given. 1.42 Example The automorphism group of the Fano plane. (0 1 2 3 4 5 6) (0 3) (1) (2 6 4 5) (0 3 1) (2) (4 5 6) (0) (1 3) (2) (6) (4 5) The four permutations on the left generate a permu- tation group Γ of order 168. Under the action of Γ, the orbit of {0, 1, 3} contains 7 triples; that of {0, 1, 5} contains seven (different) triples; and that of {0, 1, 2} contains the remaining 21 triples. 1.43 Remark Example 1.42 constructs the recurring example via the action of a group; indeed the orbit of {0, 1, 3} under Γ provides the seven triples. As there are no other group actions that preserve this orbit, Γ is the full automorphism group of the design. See §VII.9 for relevant descriptions of groups. In this representation, many symmetries are provided, not all of which are easily seen geometrically. One particular symmetry of interest is that the seven points lie in a single orbit (point transitivity); a second is that all blocks lie in the same orbit (block transitivity). See §VII.5. 1.44 Example Begin again with the 3-set {1, 2, 4} in Z7 from Example 1.40. Calculate differences modulo 7 among these three elements to obtain ±1, ±2, and ±3, all nonzero differences in Z7. 1.45 Remark In Example 1.42, not only is there a transitive action on the seven points, the action is a single 7-cycle. In Example 1.44, this is seen by the occurrence of every nonzero difference among the points once. Such cyclic designs have received particular attention. See §II.18 for cyclic and transitive symmetric designs (difference sets) and §VI.16 for cyclic block designs (difference families); also see §V.5 and §VI.17 for generalizations of orthogonal arrays defined by group actions. 1.3 Parallelism 1.46 Example On {0, 1, 2, 3,4, 5, 6}× {0, 1, 2}, form the 70 sets B = {{(i, 0), (i, 1), (i, 2)} : i = 0, . . ., 6} ∪ {{(i + 1, j), (i + 2, j), (i + 4, j)} : i = 0, . . ., 6, j = 0, 1, 2} ∪ {{(i + 3, a), (i + 5, b), (i + 6, c)} : i = 0, . . ., 6, {a, b, c} = {0, 1, 2}}. Arithmetic in the first component is done modulo 7, and in the second modulo 3. 1.47 Remark Example 1.46 contains three copies of Example 1.31 within it; they lie on the points {0, 1, 2, 3, 4,5,6} × {j} for j ∈ {0, 1, 2}. Every two of the 21 points occur
- 41. I.1.4 Closing 11 together in exactly one of the 70 sets. This is a recursive construction of a larger design from a smaller one. Now rewrite Example 1.46. 1.48 Example For i ∈ {0, 1, 2, 3, 4,5,6} let Pi = {{(i, 0), (i, 1), (i, 2)}, {(i + 1, 0), (i + 2, 0), (i + 4, 0)}, {(i + 3, 0), (i + 5, 1), (i + 6, 2)}, {(i + 1, 1), (i + 2, 1), (i + 4, 1)}, {(i + 3, 1), (i + 5, 2), (i + 6, 0)}, {(i + 1, 2), (i + 2, 2), (i + 4, 2)}, {(i + 3, 2), (i + 5, 0), (i + 6, 1)}}. For j ∈ {0, 1, 2} let Qj = {{(i + 3, j), (i + 5, j + 2), (i + 6, j + 1)} : i = 0, . . ., 6}. The 70 sets in B are partitioned into ten classes (P0, P1, . . ., P6, Q1, Q2, Q3) of seven sets each in this manner. 1.49 Remark In Example 1.48, consider one of the classes in the partition. It contains seven sets each of size 3, chosen from a set system on 21 points. Asking again for “balance,” one would hope that every point occurs in exactly one set of the class. And indeed it does! Each class contains every point exactly once; it is a parallel class or a resolution class. The whole design is partitioned into resolution classes; it is resolvable. The partition into classes is a resolution. Geometrically, one might think of the sets as lines, and a resolution class as a spanning set of parallel lines. Then resolvability is interpreted as a “parallelism.” Resolvability is a key concept in combinatorial design theory. See §II.7 and §IV.5. Indeed resolvability also arises extensively in tournament scheduling (§VI.3,§VI.51). 1.4 Closing 1.50 Remark This short introduction has afforded only a cursory glance at the many ways in which such simple objects are generated, classified, enumerated, generalized, and applied. The main themes of combinatorial design theory, balance and symmetry, recur throughout the handbook. Nevertheless, there is more to learn from the initial tour. Often the representation chosen dictates the questions asked; one seemingly simple object appears in a myriad of disguises. To explore this world further, one piece of advice is in order. Be mindful of the ease with which any combinatorial design can arise in an equivalent formulation, with different vernacular and different lines of research. The intricate web of connections among designs may not be a blessing to those entering the area, but it is nonetheless a strength. 2 Design Theory: Antiquity to 1950 Ian Anderson Charles J. Colbourn Jeffrey H. Dinitz Terry S. Griggs Those who do not read and understand history are doomed to repeat it. (Harry Truman) At the time of writing, every passing year sees the addition of more than five hundred new published papers on combinatorial designs and likely thousands more employing the results and techniques of combinatorial design theory. The roots of modern design theory are diverse and often unexpected. Here a brief history is given. It is intended to provide first steps in tracing the evolution of ideas in the field.
- 42. 12 Design Theory: Antiquity to 1950 I.2 The literature on latin squares goes back at least 300 years to the monograph Koo-Soo-Ryak by Choi Seok-Jeong (1646–1715); he uses orthogonal latin squares of order 9 to construct a magic square and notes that he cannot find orthogonal latin squares of order 10. It is unlikely that this was the first appearance of latin squares. Ahrens [68] remarks that latin square amulets go back to medieval Islam (c1200), and a magic square of al-Buni, c 1200, indicates knowledge of two 4 × 4 orthogonal latin squares. In 1723, a new edition of Ozanam’s four-volume treatise [1712] presented a card puzzle that is equivalent to finding two orthogonal latin squares of order 4. Then in 1776, Euler presented a paper (De Quadratis Magicis) to the Academy of Sciences in St. Petersburg in which he again constructed magic squares of orders 3, 4, and 5 from orthogonal latin squares. He posed the question for order 6, now known as Euler’s 36 Officers Problem. Euler was unable to find a solution and wrote a more extensive paper [798] in 1779/1782. He conjectured that no solution exists for order 6. Indeed he conjectured further that there exist orthogonal latin squares of all orders n except when n ≡ 2 (mod 4): et je n’ai pas hésité d’en conclure qu’on ne saurait produire aucun quarré complet de 36 cases, et que la même impossibilité s’étende aux cas de n = 10, n = 14 et en général à tous les nombres impairement pairs. No progress was to be made on this problem for over a century, although it was not neglected by mathematicians of the day. Indeed, Gauss and Schumacher (see [862]) corresponded in 1842 about work of Clausen establishing the impossibility when n = 6 and conjecturing (independently of Euler) the impossibility when n ≡ 2 (mod 4), which work was apparently not published. In 1900, by an exhaustive search, Tarry [2004] showed that none exists for n = 6, and in 1934 the statisticians Fisher and Yates [820] gave a simpler proof. Petersen [1731] in 1901 and Wernicke [2129] in 1910 published fallacious proofs of Euler’s conjecture for n ≡ 2 (mod 4). Then in 1922 MacNeish [1499] published another, after pointing out the error in Wernicke’s approach! In 1942 Mann [1520] described the construction of orthogonal latin squares using orthomorphisms. The falsity of the general Euler conjecture was finally estab- lished in 1958 when Bose and Shrikhande [308] constructed two orthogonal squares of order 22; and then in 1960 Bose, Shrikhande, and Parker [309] established that two orthogonal latin squares of order n exist for all n ≡ 2 (mod 4), other than 2 and 6. The study of block designs can be traced back to 1835, when Plücker [1754], in a study of algebraic curves, encountered a Steiner triple system of order 9, and claimed that an STS(m) could exist only when m ≡ 3 (mod 6). In 1839 he correctly revised this condition to m ≡ 1, 3 (mod 6) [1755]. Plücker’s discovery that every nonsingular cubic has 9 points of inflection, defining 12 lines each of 3 points, and with each pair of points on a line, illustrates the early connection between designs and geometry. Another discovery of Plücker concerned the double tangents of fourth order curves, namely, a design of 819 4-element subsets of a 28-element set such that every 3-element subset is in exactly one of the 4-element subsets, a 3-(28, 4, 1) design. Plücker raised the question of constructing in general a t-(m, t + 1, 1) design. In England, a prize question of Woolhouse [2167] in the Lady’s and Gentleman’s Diary of 1844 asked: determine the number of combinations that can be made out of n symbols, p symbols in each; with this limitation, that no combination of q symbols, which may appear in any one of them shall be repeated in any other. In 1847 Kirkman [1300] dealt with the existence of such a system in the case p = 3 and q = 2, constructing STS(m) for all m ≡ 1, 3 (mod 6). The next case to be solved, p = 4 and q = 3, was done more than a century later by Hanani [1035] in 1960. The reason the triad systems (as Kirkman called them) are not now called Kirkman triple
- 43. I.2 Design Theory: Antiquity to 1950 13 systems but rather Steiner triple systems lies in the fact that Steiner [1956], apparently unaware of Kirkman’s work, asked about their existence in 1853; subsequently Reiss [1799] (again) settled their existence, and later Witt [2159] named the systems after Steiner. A particular irony is that Steiner had actually asked a related but different question, which was taken up by Bussey in 1914 [407]. The name Kirkman has instead come to be attached to resolvable Steiner systems, on account of Kirkman’s famous 15 schoolgirls problem [1303]: fifteen young ladies in a school walk out three abreast for seven days in succession; it is required to arrange them daily, so that no two shall walk twice abreast. The first solution to the schoolgirls problem to appear in print was by Cayley [442] in 1850 followed in the same year by another by Kirkman [1302] himself. Kirkman’s so- lution involved an ingenious combination of an STS(7) and a (noncyclic) Room square of side 7. In 1863, Cayley [443] pointed out that his solution could also be presented in this way. Cayley’s and Kirkman’s solutions are nonisomorphic as Kirkman triple systems but isomorphic as Steiner triple systems, being the two different resolutions into parallel classes of the point-line design of the projective geometry PG(3, 2). In 1860 Peirce [1726] found all three solutions of the 15 schoolgirls problem having an automorphism of order 7, and in 1897 Davis [634] published an elegant geometric solu- tion based on representing the girls as the 8 corners, the 6 midpoints of the faces, and the centre of a cube. In 1912, Eckenstein [769] published a bibliography of Kirkman’s schoolgirl problem, consisting of 48 papers. The 7 nonisomorphic solutions of the 15 schoolgirls problem were enumerated by Mulder [1644] and Cole [593]. The first pub- lished solution of the existence of a Kirkman triple system of order m, KTS(m), for all m ≡ 3 (mod 6) was by Ray-Chaudhuri and Wilson [1781] in 1971. The problem had in fact already been solved by the Chinese mathematician Lu Jiaxi at least eight years previously, but the solution had remained unpublished because of the political upheavals of the time. Lu also solved the corresponding problem for blocks of size 4; these results were eventually published in his collected works [1484] in 1990. Prior claim to the authorship of the 15 schoolgirls problem was made by Sylvester [1996] in 1861, hotly refuted by both Kirkman [1305] and Woolhouse [2168]. But Sylvester raised further questions about such systems, for example: can all 455 triples from a 15-element set be arranged into 13 disjoint KTS(15)s, thereby allowing a walk for each of the 13 weeks of a school term, without any three girls walking together twice? The problem was finally solved in 1974 by Denniston [690] although both Kirkman and Sylvester had solutions for the corresponding problem with 9 instead of 15 (indeed they had the first example of a large set of designs). Surely neither Sylvester nor Denniston would have anticipated that Denniston’s solution would form the basis in 2005 of a musical score, Kirkman’s Ladies [1212], about which the composer states: The music seems to come from some point where precise organization meets near chaos. The problem of the existence of large sets of KTS(n) in the general case of n ≡ 3 (mod 6) is still unsolved. Sylvester also asked about the existence of a large set of 13 disjoint STS(15)s. In this case, the general problem of large sets of STS(n) is solved for all n ≡ 1, 3 (mod 6), n 6= 7, mainly by work of Lu [1482, 1483] but also Teirlinck [2015]. Cayley [442] had shown that the maximum number of disjoint STS(7)s on the same set is 2. In 1917, Bays [169] determined that there are precisely 2 nonisomorphic large sets of STS(9). Kirkman’s solution of the 15 schoolgirls problem was extended brilliantly in 1852– 3 by Anstice [103, 104]; he generalised the case where n = 15 to any integer of the form n = 2p + 1, where p is prime, p ≡ 1 (mod 6). For the first time making use of primitive roots in the construction of designs (a method basic to Bose’s 1939 paper) and introducing the method of difference families, Anstice constructed an infinite class
- 44. 14 Design Theory: Antiquity to 1950 I.2 of cyclic Room squares, cyclic Steiner triple systems, and 2-rotational Kirkman triple systems. His starter-adder approach to the construction of Room squares includes the Mullin–Nemeth starters [1653] as a special case. In 1857 Kirkman [1304] used difference sets (in the equivalent formulation of perfect partitions) to construct cyclic finite projective planes of orders 2, 3, 4, 5, and 8, introducing at the same time the concept of a multiplier of a difference set. He also wrote that he thought that a solution for 6 is “improbable,” for 7 is “very likely” (because 7 is prime) and that he did not see why one should not be discovered for 9. He did not comment on order 10. A few years earlier, Kirkman [1301] had shown how to construct affine planes of prime order, and, hence, essentially by adding points at infinity, finite projective planes of all prime orders (although he did not use the geometric terminology). This paper also showed how to construct certain families of pairwise balanced designs. Such designs were to prove invaluable in later combinatorial constructions; indeed, in the same paper Kirkman himself used them to solve the schoolgirls problem for 5 · 3m+1 symbols. Kirkman’s work on combinatorial designs is indeed seminal. To quote Biggs [269]: Kirkman has established an incontestable claim to be regarded as the founding father of the theory of designs. Among his contemporaries, only Sylvester attempted anything com- parable, and his papers on Tactic seem to be more concerned with advancing his claims to have discovered the subject than with advancing the subject itself. Not until the Tactical Memoranda of E. H. Moore in 1896 is there another contribution to rival Kirkman’s. In 1867, Sylvester [1997] investigated a tiling problem and constructed what he called anallagmatic pavements, which are equivalent to Hadamard matrices. He showed how to construct such of order 2n from a solution of order n; twenty-six years later Hadamard [1003] showed that Hadamard matrices give the largest possible determi- nant for a matrix whose entries are bounded by 1. Hadamard showed that the order of such a matrix had to be 1, 2, or a multiple of 4, and he constructed matrices of orders 12 and 20. In 1898 Scarpis [1846] showed how to construct Hadamard matrices of order 2k · p(p + 1) whenever p is a prime for which a Hadamard matrix of order p + 1 exists. In 1933 Paley [1713] used squares in Fq to construct Hadamard matrices of order q + 1 when q ≡ 3 (mod 4) and of order 2(q + 1) when q ≡ 1 (mod 4). He also showed that if m = 2n , then the 2m (±1)-sequences of length m can be parti- tioned in such a way as to form the rows of 2m−n Hadamard matrices. In a footnote Paley acknowledged that some of his results had been announced by Gilman in the 1931 Bulletin of the AMS, but without proof. In the same journal issue in 1933, Todd [2036] pointed out the connection between Hadamard matrices and Hadamard designs; designs with these parameters were relevant to work of Coxeter (also in the same journal issue) on regular compound polytopes! At this time the matrices were not yet called Hadamard matrices; Paley called them U-matrices. Williamson [2142] first called them Hadamard matrices in 1944. The existence of Hadamard matrices for all admissible orders remains open. Cayley introduced the word tactic for the general area of designs. The word config- uration, used nowadays more generally, was given a specific meaning in 1876 by Reye [1800] in terms of geometrical structures. Essentially, a configuration was defined to be a system of v points and b lines with k points on each line and r lines through each point, with at most one line through any two points (and hence any two lines intersecting in at most one point). If v = b (and therefore k = r), the configuration is symmetric and denoted by vk; thus, for example, the Fano plane is 73, the Pappus con- figuration is 93, and the Desargues configuration is 103. In fact, the first appearance of the Petersen graph is not in Petersen’s paper of 1891 but in 1886 by Kempe [1277] as the graph of the Desargues configuration. In 1881, Kantor [1250] showed that there exist one 83, three 93, and ten 103 configurations. Six years later, Martinetti [1528]
- 45. I.2 Design Theory: Antiquity to 1950 15 showed there were thirty-one 113 configurations. In the following years, Martinetti constructed symmetric 2-configurations (two points are connected by at most 2 lines), with parameters (74)2, (84)2, and (94)2 as well as the biplanes (115)2 and (166)2. The biplane (115)2, also a Hadamard design 2-(11, 5, 2), had previously been constructed by Kirkman [1305] in 1862. Symmetric configurations v4 were first studied in 1913 by Merlin [1593]. In 1942, Levi [1442] unified the treatment of configurations with questions in algebra, geometry, and design theory. Interest in counting configurations is reflected in the enumerative work on Steiner systems. It was known early on that there are unique STS(7) and STS(9). In 1897, Zulauf [2219] showed that the known STS(13)s fall into two isomorphism classes. Using the connection between STS(13)s and symmetric configurations 103, in 1899 De Pasquale [659] determined that only two isomorphism classes are possible. The same result was also obtained two years later by Brunel. The enumeration of nonisomorphic STS(15)s was first done by Cole, Cummings, and White. In 1913, both Cole and White published papers introducing ideas for enumeration and a year later Cummings classified all 23 STS(15)s which have a subsystem of order 7. Then in 1919, White, Cole, and Cummings [2135] in a remarkable memoir, succeeded in determining that there are precisely 80 nonisomorphic STS(15)s. In 1940, Fisher [819], unaware of this catalogue, also generated STS(15)s; he used trades on Pasch configurations to find 79 of the 80 systems. The veracity of White, Cole, and Cummings’ work was confirmed in 1955 by Hall and Swift [1020] in one of the first cases in which digital computers were used to catalogue combinatorial designs. The number of nonisomorphic STS(19)s was published only in 2004 [1268], perhaps understandably as it is 11,084,874,829. Enumeration of latin squares has a more complex history (see [1573]); early results are by Euler [798] in 1782, Cayley [446] and Frolov [834] in 1890, Tarry [2004] in 1900, MacMahon [1498] in 1915, Schönhardt [1853] in 1930, Fisher and Yates [820] in 1934, and Sade [1836] and Saxena [1844] in 1951. By the end of the 19th century there was no formal body of work called graph theory, but some graph theoretic results were in existence in the language of config- urations or designs. One-factorizations of K2n are resolvable 2-(2n, 2, 1) designs, and the “classical” method of construction appeared in Lucas’ 1883 book [1488]; the con- struction was credited to Walecki and can be described in terms of chords of a circle or in terms of a difference family now known as a patterned starter. Remarkably, in 1847 Kirkman [1300] already gave a construction based on a lexicographic method (in fact the greedy algorithm), which gives a schedule isomorphic to that of Lucas– Walecki. Lucas’ book also describes the construction of round-dances (or hamiltonian decompositions of the complete graph) by means of what we now call a terrace. Enu- meration results for nonisomorphic 1-factorizations of K2n span nearly a century. In 1896, Moore [1626] reported that there are 6 distinct 1-factorizations of K6 that fall into a single isomorphism class. The 6 nonisomorphic 1-factorizations of K8 appear in Dickson and Safford [703]. Gelling [887] showed that the number of nonisomorphic 1-factorizations of K10 is 396, and Dinitz, Garnick, and McKay [715] showed that there are 526,915,620 nonisomorphic 1-factorizations of K12. In 1891, Netto [1670], unaware of Kirkman’s work, gave four constructions for Steiner triple systems: (i) an STS(2n+1) from an STS(n), (already used by Kirkman), (ii) an STS(mn) from an STS(m) and an STS(n), (iii) an STS(p) where p is a prime of the form 6m+1, and (iv) an STS(3p) where p is of the form 6m+5. The construction given by Netto for (iii) involves primitive roots, but is not the construction usually associated with his name. These constructions enabled Netto to construct STS(n) for all admissible n < 100 except for n ∈ {25, 85}. In 1893 Moore [1624] completed Netto’s proof, giving a construction of an STS(w + u(v − w)) from an STS(u) and an STS(v) with an STS(w) subsystem. In [1624] he also proved that for all admissible
- 46. 16 Design Theory: Antiquity to 1950 I.2 v > 13, there exist at least two nonisomorphic STS(v). In 1895 and 1899, Moore [1625, 1627] determined the automorphism groups of the unique STS(7) and 3-(8, 4, 1) designs and studied the intersection properties of the 30 realizations on the same base set of both systems. Undoubtedly the most far-reaching paper by Moore is his “Tactical Memoranda I–III” [1626]. This was published in 1896 in the American Journal of Mathematics (founded in 1878 by Sylvester during his seven-year stay in the U.S. as professor at Johns Hopkins University). It contains many jewels hidden in 40 pages of difficult and often bewildering terminology and notation. After a brief introductory section (memorandum I), memorandum II introduces a plethora of different types of designs, including with order and/or with repeated block elements, the first occurrence of both of these concepts. Here we find the construction using finite fields of a complete set of q − 1 mutually orthogonal latin squares (MOLS) of order q whenever q is a prime power (a result rediscovered much later by Bose [300] in 1938 and by Stevens [1961] in 1939 independently). One also finds the theorem, later rediscovered by MacNeish [1499], that if there exist t MOLS of order m and of order n, then there exist t MOLS of order mn. Further, it contains the construction of many 1-rotational designs including Kirkman triple systems and work on orthogonal arrays. Memorandum III contains the first general results on whist tournaments and triple-whist tournaments. Whist tournaments Wh(4n) had been presented by Mitchell [1618] in 1891 for several small values of n; Moore showed how to construct several infinite families of such designs. These are resolvable 2-(4n, 4, 3) designs with extra properties, and are the first examples of nested designs to appear in the literature. Further in this section, Moore points out the relationship between whist tournaments Wh(4n) and resolvable 2-(4n, 4, 1) designs and gives a cyclic construction of such designs in the case where 4n = 3p + 1, where p is a prime of the form 4m + 1. At this time much work was being done on finite geometries and tactical configurations. For example, Fano [807] described a number of finite geometries in 1892, and in 1906 Veblen and Bussey [2097] used finite projective geometries to construct many designs, in particular finite projective planes of all prime power orders. Triple systems were also studied by Heffter whose approach to the subject stemmed from his study of triangulations of complete graphs on orientable surfaces, relating these to twofold triple systems. In 1891 he gave a particularly simple construction for twofold triple systems of order 12s + 7 [1076]. Although Kirkman, nearly 40 years earlier, had pointed to the existence of an indecomposable 2-(7, 3, 3) design without repeated blocks, this seems to be the first known infinite class of designs with λ > 1. In 1896, Heffter [1077] introduced his famous first difference problem, in relation to the construction of cyclic STS(6s + 1), and a year later both the first and second difference problems appeared [1078]. Heffter’s difference problems were eventually solved in 1939 by Peltesohn [1727], and later a particularly elegant approach by Skolem [1924, 1925] was given in 1957– 58. Skolem’s interest in Steiner triple systems was longstanding. In 1927 he wrote notes for the second edition of Netto’s book [1671], whose first edition appeared in 1901; there he constructed STS(v) for all products of primes of the form 6n + 1 and showed that if v = 6s + 1 then there are at least 2s such STS(v). Then in [1924], he introduced the idea of a pure Skolem sequence of order n and proved that these exist if and only if n ≡ 0, 1 (mod 4). In [1925] he extended this idea to that of a hooked Skolem sequence, the existence of which for all admissible n, along with that of pure Skolem sequences, would constitute a complete solution to Heffter’s first difference problem. O’Keefe [1695] proved that a hooked Skolem sequence exists if and only if n ≡ 2, 3 (mod 4). In 1923–34, Bays [170, 171] undertook a systematic study of group actions on
- 47. I.2 Design Theory: Antiquity to 1950 17 cyclic STS, and also in [172] on SQS; this culminated in a remarkable theorem, the Bays–Lambossy theorem [1382] demonstrating that isomorphism of cyclic structures of prime orders is the same as multiplier equivalence. The period between the two world wars saw a major step forward in the study of statistical experimental design. Already by 1788, de Palluel [618] had used a 4 × 4 latin square to define an experimental layout for wintering sheep (see [1985] for a modern treatment of his work). In 1924 Knut Vik [2100] described the use of certain latin squares in field experiments. But in 1926 [818], Fisher, at Rothamsted Experimental Station, indicated how orthogonal latin squares could be used in the construction of experiments. This led to detailed work in listing and enumerating all latin squares of small orders; in particular, in 1934 Fisher and Yates [820] (Fisher’s successor at Rothamsted when he left to take up a chair in London) enumerated all 6 × 6 latin squares and established that no two were orthogonal. Then, in a 1935 paper delivered to the Royal Statistical Society, Yates [2180] drew attention to the importance of balanced block designs for statistical design. In [2181] Yates called them symmetrical incomplete randomized blocks, the present terminology of balanced incomplete block designs being introduced by Bose in 1939; but the present use of v, b, r, k, and λ goes back to Yates (except that Yates originally used t for treatment instead of v for variety). It was also in 1935 that Yates [2179] gave the first formal treatment of the factorial designs proposed by Fisher in 1926. In 1938, Fisher and Yates [821] published their important book Statistical Tables for Biological, Agricultural and Medical Research which, along with much statistical data, presented what was known about latin squares of small order, including complete sets of orthogonal latin squares of orders 3, 4, 5, 7, 8, and 9, as well as tables of balanced incomplete block designs with replication number r up to 10. Some gaps would be filled by Bose the following year, and some gaps would be ruled out by Fisher’s inequality. The years 1938–39 saw the publication of several important papers. Peltesohn [1727] gave a complete solution to Heffter’s difference problems and hence established the existence of cyclic Steiner triple systems for all admissible orders with the exception of v = 9. Singer [1920] used hyperplanes in PG(2, q) to construct cyclic difference sets, thereby establishing the existence of cyclic finite projective planes (a few of which had been found by Kirkman) for all prime power orders. Singer’s cyclic difference sets gave the first important family of difference sets apart from those implicitly in the work of Paley, although many examples of difference families had been used by Anstice and Netto. In 1942 Bose [302] gave an affine analogue of Singer’s theorem. Here he gave the first examples of relative difference sets and suggested the notion of a group divisible design (GDD). Chowla [501] constructed difference sets using biquadratic residues in 1944. In 1947, Hall [1012] published an important paper on cyclic projective planes. Difference sets were studied in depth after World War II; the important multiplier theorem of Hall and Ryser [1019] dates from 1951. In 1938, there was also the remarkable paper by Witt [2159] on Steiner systems, detailing the existence of several Steiner systems with t = 3, including various families obtained geometrically and the Steiner systems S(3, 4, 26) and S(3, 4, 34) constructed by Fitting [822] in 1914. But of greater interest, the paper also includes the systems S(5, 8, 24) and S(5, 6, 12) with the Mathieu groups M24 and M12, respectively, as their automorphism groups. Witt gives a construction of these systems as a single orbit of a starter block under the action of the groups PSL2(23) and PSL2(11), respectively, and proofs of the uniqueness of the designs as well as of their derived subsystems. He also gives a proof of the uniqueness of the system S(3, 5, 17) and of the nonexistence of a system S(4, 6, 18). The results on the Mathieu systems had already appeared in 1931 by Carmichael [437, 438]. However in 1868 in the Educational Times, Lea [1418] both proposed the
- 48. 18 Design Theory: Antiquity to 1950 I.2 problem and gave a solution to constructing the system S(4, 5, 11). The next year another solution appeared, this time by none other than Kirkman [1306], who went on to try to construct a system S(4, 5, 15). The fact that this system does not exist was not shown until 1972 by Mendelsohn and Hung [1588]. Because it is relatively easy to construct the system S(5, 6, 12) from the S(4, 5, 11), it is perhaps a surprise that Kirkman did not do so. Instead, this was left to Barrau [161] in 1908 who, using purely combinatorial methods, proved the existence and uniqueness of both the systems S(4, 5, 11) and S(5, 6, 12). The work of Fisher and Yates created growing interest in designs among statisti- cians, and in 1939 Bose [301], in Calcutta, published a major paper on the subject. To quote Colbourn and Rosa [578]: Sixty years on, the paper of Bose forms a watershed. Before then, while combinatorial design theory arose with some frequency in other researches, it was an area without a basic theme, and without substantial application. After the paper of Bose, the themes of the area that recur today were clarified, and the importance of constructing designs went far beyond either recreational interest, or as a secondary tactical problem in a larger algebraic or geometric study. Bose presents the study of balanced incomplete block designs as a coherent theory, using finite fields, finite projective geometries, and difference methods to create many families of designs that contain many old and many new examples. There is much material on designs with λ > 1. The use of difference families, both pure and mixed, carried on the work of Anstice of which he was unaware. Bose’s paper appeared in volume 9 of the Annals of Eugenics. It is remarkable what combinatorial design theory appeared in that volume of 1939. Stevens [1961] discusses the existence of complete sets of orthogonal latin squares; Savur [1843] gives (yet) another construction of STSs; and Norton [1686] discusses 7×7 latin and graeco-latin squares. Stevens and Norton had presented their work at the 1938 meeting of the British Association in Cambridge, where Youden [2195] introduced the experimental designs subsequently known as Youden squares. These arrays are equivalent to latin rectangles whose columns are the blocks of a symmetric balanced design. That this representation is always possible is a consequence of Philip Hall’s 1935 theorem on systems of distinct representatives (or, equivalently, König’s theorem), widely known as the Marriage Problem, which has had many applications in the study of designs. In 1938, Bose [300] proved that the existence of a complete set of n − 1 MOLS of order n is equivalent to the existence of a finite projective plane of order n. Then in 1938, Bose and Nair [306] introduced the ideas of association schemes and partially balanced incomplete block designs. In 1942, Bose [303] gave nontrivial examples of balanced incomplete block designs with repeated blocks. The term association scheme was introduced by Bose and Shimamoto [307] in 1952. Bose moved to the US in 1949, and later with Shrikhande and Parker established the falsity of the Euler conjecture, as mentioned above. This work showed clearly the usefulness of pairwise balanced designs. While Bose’s 1939 paper exploits the algebra of finite fields, the connection with algebra explored at the time was much broader. In 1877, Cayley [444, 445] discussed the structure of the multiplication table of a group (its Cayley table). Schröder, from 1890 through 1905, published a two-thousand page treatise [1857] on algebras with a binary operation, and extensively discussed quasigroups with various restrictions. Frolov [834] and Schönhardt [1853] exploited the connection between latin squares and quasigroups. However, only in the 1930s and 1940s did the area take form. In 1935, Moufang [1640] studied a specific class of quasigroups, introducing in the process the Moufang loops. In 1937, Ore and Hausmann [1701] first used the term quasigroup in
- 49. Discovering Diverse Content Through Random Scribd Documents
- 50. listening to these gentlemen, who had for a long time forgotten the existence of the Statute, even as a simple historical document,— (Laughter.)—that the Statute runs a serious risk and that one cannot even discuss nor examine it. Well, I think that none of you can consider Camillo Cavour as a Bolshevist and a Fascista of 1848. Everybody knows that the Constitutional movement of Piedmont was the work of Cavour. Everybody knows how the political Constitution was granted. At Genoa a tumult arose against the Jesuits, believed supporters of Absolutism. A Commission of Genoese went to Turin and asked for the expulsion of the Jesuits and the calling out of the Civic Guard. But Cavour answered: “This is too little, the times are ripe for something more!” Cavour wrote in his paper, Il Risorgimento: “The Constitution must be demanded.” And this was promulgated on the 4th of March. In its preamble it says: “The Statute is the fundamental, perpetual law of the Monarchy.” Four days afterwards the first Constitutional Ministry of Coalition was formed with the Moderate Balbo and the Democratic Pareto. The phrase “The Statute is the fundamental, perpetual and irrevocable law of the Monarchy” had wounded the ears of the Democrats. Cavour hastened to interpret it in a relative sense. It is worth while to listen attentively to this paragraph of Cavour. “How is it possible,” he said, “how can it be expected that the legislator would have wished to pledge himself and the nation not to make the slightest direct change, to bring the smallest improvement to a political law? But this would mean the removal from the community of the power of revising the Constitution; it would mean the deprival of the indispensable power of modifying its political form according to new social exigencies; this would be such an absurd idea that no one of those who co-operated in the making of this fundamental law could conceive it. A nation cannot renounce the power of changing by legal means its common law.” After a short time history had to register a first violation of the Statute, which assumed or presumed that, in order to become a member of Parliament, it was necessary to be an Italian citizen. On
- 51. the 16th of October there was a division between the Right, amongst which there were the Moderates and the Municipals, and the Left, to which belonged the Democrats, called the “burnt heads,” and the Republicans. On the following day these two parties were agreed in unanimously proclaiming above the Statute that all Italians could belong to the Subalpine Parliament. The first to benefit by this violation of the Statute was Alessandro Manzoni; but he declined the mandate by a letter which represents a fine example of correctness and political probity. (Approval.) Nobody, Gentlemen, wishes to overthrow or destroy the Statute, which rests solidly on firm foundations; but the inhabitants of this building from 1848 up to to-day have changed. There are other exigencies, other needs. There is no longer the Piedmontese Italy of 1848! And it is very strange to notice among the defenders of the Statute those who have violated it in its fundamental laws, those who have curtailed the prerogatives of the Crown, those who wanted the Crown to be entirely outside the politics of the nation, and to become a dead institution. (Loud applause.) The Abolition of Parliament? They say that this Government does not like the Chamber of Deputies. (Comments.) They say that we want to abolish Parliament and deprive it of all its essential attributes. It is timely to say that the collapse of Parliament is not desired by me, nor by those who follow my ideas. Parliamentarism has been severely affected by two phenomena typical of our days: on one side Syndicalism, on the other Journalism. Syndicalism gathers by its various organisations all those who have special interests to protect, who wish to withdraw them from the manifest incompetence of the political Assembly. Journalism represents the daily Parliament, the daily platform where men coming from the Universities, from Science, Industry, from the experience of life itself, dissect problems with a competence that is very seldom found on the Parliamentary benches.
- 52. These two phenomena typical of the last period of capitalist civilisation are those which have reduced the enormous importance which was attributed to Parliament. To sum up, Parliament can no longer contain all the life of the nations, because modern life is exceptionally complicated and difficult. But this does not mean that we wish to abolish Parliament. We wish rather to improve it, to make it more perfect, make it a serious, if possible a solemn institution. In fact, if I had wished to abolish Parliament, I would not have introduced an Electoral Reform Bill. This Bill logically presupposes the elections, and through these elections there will be deputies—(Laughter.)—who will form Parliament. In 1924, therefore, there will be a Parliament. But must the Government be towed along by Parliament? Must it be at the mercy of Parliament? Must it be without a will, or a head before Parliament? I cannot admit that. The Great Fascista Council. They say that Fascismo has created duplicate institutions. These duplicates do not exist. The Great Fascista Council is not a duplicate of the Council of Ministers or above it. It met four times and never dealt with problems which concerned the Council of Ministers. With what, then, did the Great Fascista Council deal? In the February meeting it devoted itself to the National Militia and Freemasonry; it paid a tribute to the Dalmatians and to the people of Fiume, and dealt with Fascismo abroad. In the March meeting it arranged the ceremony for the anniversary of the foundation of Rome and dealt with Syndicalism. In its fourth meeting it devoted itself to the Congress of Turin and again to Syndicalism. All the great problems dealing with State administration, with the reorganisation of armed forces, with the reform of our judiciary circuits, with the reform of the schools, all the measures of a financial nature have been adopted directly by the responsible body, the Council of Ministers.
- 53. And then what is the Great Fascista Council? It is the organ of co- ordination between the responsible forces of the Government and those of Fascismo. Among all the organisations created after the October revolution, the Great Fascista Council is the most characteristic, the most useful, the most efficient. I have abolished the High Commissioners, because they duplicated the Prefects and also embarrassed the authority of the latter, who alone have the right to wield authority. But I could never think of abolishing the Great Fascista Council, not even if to-morrow by chance the Council of Ministers were composed entirely of Fascisti. Our Magnanimity must not be taken advantage of! This Government, which is depicted as hostile to liberty, has been perhaps too generous. The October revolution has not been bloodless for us; we have left dozens and dozens of dead. And who would have prevented us from doing in those days that which all revolutions have done, from freeing ourselves once for all from those who, taking advantage of our magnanimity, now render our task difficult? Only the Socialists of the newspaper La Giustizia, of Milan, have had the courage to recognise that if they still exist they owe it to us, who did not wish that, in the first moments of “The March on Rome,” the “black shirts” should be stained with Italian blood. But our generosity must not be taken advantage of! Nobody must hope for a Crisis in Fascismo. The Membership of Fascismo. But nobody must hope for a crisis in Fascismo, which is and will remain simply a formidable party. If you happen to notice that in one of its innumerable sections in Italy there is dissension, do not thus draw the conclusion that Fascismo is in a state of crisis. When a party holds the Government in its hands it holds it, if it wishes to hold it, because it possesses formidable forces to use to consolidate its power with increasing strength. Fascismo is a Syndicalist movement which includes one million and a half of workmen and contadini, who, I must say in their praise, are those
- 54. who give me no trouble. There is, moreover, a political body which has 550,000 members, and I have asked to be relieved of at least 150,000 of these gentlemen. (Laughter.) There is, still, a military section of 300,000 “black shirts,” who are only waiting to be called. These bodies are all united by a kind of moral cement, which might be called mystic and holy, and through which, by touching certain keys, we would hear to-morrow the sounds of certain trumpets! The Associations which are included in Fascismo. They ask us: “Will you then camp out in Italy as an army of enemies which oppress the remainder of the population?” Here we have the philosophy of force by consent. In the meanwhile I have the pleasure to announce that imposing masses of men who deserve all the respect of the nation have joined Fascismo, such as the Association of the Maimed and the Disabled, the National Association of Ex-soldiers. In the wake of Fascismo, moreover, are also included the families of the fallen in war. There are a great many members coming from the people in these three Associations, whilst there is a great solidarity amongst these disabled ex-soldiers and families of the fallen in war. They represent millions of people, and, in the face of this collaboration, must I go and simply seek all the fragments, all the relics of the old traditional parties? Must I sell my spiritual birthright for a mess of pottage which might be offered to me by those who have followed no one in the country? (Loud assent.) No! I shall never do this. The Collaboration I welcome. If there is anybody who wishes to collaborate with me, I welcome him to my house. But if this collaborator has the air of a controlling inquisitor, or of the expectant heir, or of the man who lies in ambush, with the object of being able at a given moment to record my mistakes, then I declare that I do not want to have anything to do with this collaboration. (Bravo!) Besides, there is a moral force in all this. What was the cause after all which affected Italian life in past years? Italy was passing through
- 55. a transformation. There were never definite limits. Nobody had the courage to be what he should have been. There was the bourgeois who had Socialistic airs, there was the Socialist who had become a bourgeois up to his finger tips. The whole atmosphere was made up of half tones of uncertainty. Well, Fascismo seizes individuals by their necks and tells them: “You must be what you are. If you are a bourgeois you must remain such. You must be proud of your class, because it has given a type to the activity of the world in the nineteenth century. (Approval.) If you are a Socialist you must remain such, although facing the inevitable risk you run in that profession.” (Laughter.) Taxation and the Discipline of the Italian Population. The sight which to-day the nation offers is satisfactory, because the Government exercises a stern and, if you like to say so, a cruel policy. It is compelled to dismiss by thousands its officials, judges, officers, railway men, dock-workers; and each dismissal represents a cause of trouble, of distress, of unrest to thousands of families. The Government has been compelled to levy taxes which unavoidably hit large sections of the population. The Italian people are disciplined, silent and calm, they work and know that there is a Government which governs, and know, above all, that if this Government hits cruelly certain sections of the Italian people, it does not do so out of caprice, but from the supreme necessity of national order. The Government is One. Above this mass of people there are the restless groups of practising politicians. We must speak plainly. In Italy there were several Governments which, before the present one, always trembled before the journalist, the banker, the grand master of Freemasonry, before the head of the Popular Party, who remains more or less in the background,—(Applause.)—and it was enough for one of these ministers in partibus to knock at the door of the Government, for the Government to be struck by sudden paralysis. Well, all this is over! Many men gave themselves airs with the old
- 56. Governments; those I have not received, but have reduced them to tears. (Assent.) For the Government is one. It knows no other Government outside its own and watches attentively, because one must not sleep when one governs, one must not neglect facts, one must keep before one’s eyes all the panorama, notice all the composition and decomposition, the changes of parties and of men. Sometimes it is necessary, as a tactical measure, to be circumspect; but political strategy, at least mine, is intransigent and absolute. My only Ambition is to make the Italian People Strong, Prosperous, Great and Free. I should have finished; in fact I have finished, but I must still add something that concerns me a little personally. I do not deny to citizens what one might call the “Jus murmurandi”—the right of grumbling. (Laughter.) But one must not exaggerate, nor raise bogies, nor have one’s ears always open to dangers which do not exist. And, believe me, I do not get drunk with greatness. I would like, if it were possible, to get drunk with humility. (Approval.) I am content simply to be a Minister, nor have I ambitions which surpass the clearly defined sphere of my duties and of my responsibilities. And yet I, too, have an ambition. The more I know the Italian people, the more I bow before it. (Assent.) The more I come into deeper touch with the masses of the Italian people, the more I feel that they are really worthy of the respect of all the representatives of the nation. (Assent.) My ambition, Honourable Senators, is only one. For this it does not matter if I work fourteen or sixteen hours a day. And it would not matter if I lost my life, and I should not consider it a greater sacrifice than is due. My ambition is this: I wish to make the Italian people strong, prosperous, great and free! (The end of the speech is hailed by a frantic and delirious ovation. All the Senators rise, and the Tribune applauds loudly, whilst the great majority of the Senators go to congratulate the Hon. Mussolini.) (The sitting is adjourned.)
- 58. “AS SARDINIA HAS BEEN GREAT IN WAR, SO LIKEWISE WILL SHE BE GREAT IN PEACE” Speech delivered from the Palazzo della Prefettura at Sassari (Sardinia) on 10th June 1923. Citizens of Sassari! Proud people of Sardinia! The journey which I have made to-day is not, and should not be interpreted as, a Ministerial tour. I intended to make a pilgrimage of devotion and love to your magnificent land. I have been told that, since 1870 to to-day, this is the first time that the head of the Government addresses the people of Sassari assembled in this vast square. I deplore the fact that up to this day no Prime Minister, no Minister, has felt the elementary duty of coming here to get to know you, your needs, to come and express to you how much Italy owes you! (Applause.) For months, for years, during the long years of our bloody sacrifice and of our sacred glory, the name of Sassari, consecrated to history by the bulletins of war, has echoed in the soul of all Italy. Those who followed the magnificent effort of our race, those who steeped themselves in the filth of the trenches, young men of my generation—proud and disdainful of death—all those who bear in their heart the faith of their country, all those, O men of the Sassari Brigade, O citizens of Sassari, pay you tribute of a sign, of a testimony of infinite love. (Applause.) What does it matter if some lazy bureaucrat has not yet taken into account your needs? Sassari has already passed gloriously into history. I was grieved to-day when I was told that this town has no water. It is very sad that a city of heroes has to endure thirst. Well! I promise you that you will have water; you have the right to have it.
- 59. (Applause.) If the National Government grants to you, as it will grant, the three or four millions necessary for this purpose, it will only have accomplished its duty, because while elsewhere young men with broad shoulders worked at the lathe, the people of Sardinia fought and died in the trenches. We intend to raise up again the towns and all the land, because he who has contributed to the war is more entitled to receive in peace. A few days ago, on the anniversary of the war, I went by aeroplane to the cemeteries of the Carso. There are many of your brothers who sleep in those cemeteries the sleep which knows no awakening. I have known them, I have lived with them, I have suffered with them. They were magnificent, long-suffering, they did not complain, they endured, and when the tragic hour came for them to advance from the trenches they were the first and never asked why. (Loud applause.) The National Government which I have the honour to direct is a Government which counts upon you, and you can count upon it. It is a Government sprung forth from a double victory of the people. It cannot, therefore, be against the working classes. It comes to you so that you may tell it frankly and loyally what are your needs. You have been forgotten and neglected for too long! In Rome they hardly knew of the existence of Sardinia! But since the war has revealed you to Italy, all Italians must remember Sardinia, not only in words, but in deeds. (Loud applause.) I am delighted, I am deeply moved by the reception which you have given me. I have looked you well in the face, I have recognised that you are superb shoots of this Italian race which was great when other people were not born, of this Italian race which three times gave our civilisation to the barbarian world, of this Italian race which we wish to mould by all the struggles necessary for discipline, for work, for faith. (Applause.) I am sure that, as Sardinia has been great in war, so likewise will she be great in peace. I salute you, O magnificent sons of this
- 60. rugged, ferruginous, and so far forgotten island. I embrace all of you in spirit. It is not the head of the Government who speaks to you, it is the brother, the fellow-soldier of the trenches. Shout then with me: Long live the King! Long live Italy! Long live Sardinia! (An enthusiastic ovation greeted the last words of Mussolini.)
- 61. “MEN PASS AWAY, MAYBE GOVERNMENTS TOO, BUT ITALY LIVES AND WILL NEVER DIE” Speech delivered at Cagliari (Sardinia) on 12th June 1923, from the Palazzo della Prefettura. Citizens! Black shirts! Chivalrous people of Cagliari! Of late I have visited several towns, including those which belong to the place where I was born. Well! I wish to tell you, and this is the truth, that no town accorded me the welcome you gave me to-day. I knew that the town of Cagliari was peopled by men of strong passions, I knew that an ardent spirit of regeneration throbbed in your hearts. The cheers with which you welcomed me, the crowd crammed into the Roman amphitheatre, all this tells me that here Fascismo has deep roots. I thank you, therefore, Citizens, from the depth of my heart. I have come to Sardinia not only to know your land, as forty-eight hours would not be enough for that purpose, and still less would they be enough to examine closely your needs. I know them; statesmen have known them for the last fifty years. Those needs are already before the nation, and if up to to-day they have not yet been solved, this is due to the fact that Rome was lacking that iron will for regeneration which is the pivot, the essence of the Fascista Government’s faith in the future of our country. (Applause.) Passing through your land, I have found here a living, throbbing limb of the mother country. Truly this island of yours is the western bulwark of the nation; is like a heart of Rome set in the midst of our sea. Amongst all the impressions I have received in coming here, one has struck my heart. I was told that Sardinia, for special local reasons, was refractory to Fascismo. Here, too, there was another misunderstanding. But from to-day the cohorts and the legions, the
- 62. thousands of strong “black shirts,” the syndicates, the fasci, the whole youth of this island is there to show that Fascismo, representing an irresistible movement for the regeneration of the race, was bound to carry with it this island where the Italian race is manifested so superbly. (Applause.) I salute you, Black shirts! We saw each other in Rome and the groups coming from Sardinia were cheered in the capital. You bear in your hearts the faith which at a given moment drove thousands and thousands of Fascisti from all the cities, from all the villages of Italy, to Rome. (Applause.) Nobody can ever dream of wrenching from us the fruit of victory that we have paid for by so much blood generously shed by youths who offered their lives in order to crush Italian Bolshevism. Thousands and thousands of those who suffered martyrdom in the trenches, who have resumed the struggle after the war was over, who have won—all those have ploughed a furrow between the Italy of yesterday, of to-day and of to-morrow. Citizens of Cagliari! You must certainly play a part in this great drama. You, undoubtedly, wish to live the life of our great national community, of this our beloved Italy, of this adorable mother who is our dream, our hope, our faith, our conviction, because men pass away, maybe Governments, too, but Italy lives and will never die! (Loud applause.) To-day I have visited the marvellous works of the artificial Lake Tirso. They are not only a glory to Sardinia, they represent a masterpiece of which the whole nation may be proud. I feel, almost by intuition, that Sardinia also, too long forgotten, perhaps too patient, Sardinia to-day marches hand in hand with the rest of Italy. Let us then salute each other, O Citizens! After this speech of mine, which was meant to be an act of devotion, a bond of union between us, let us salute each other by shouting: Long live the King! (Cheers.) Long live Italy! (Cheers.) Long live Fascismo! (Loud cheers.)
- 64. “FASCISMO WILL BRING A COMPLETE REGENERATION TO YOUR LAND” Speech delivered at Iglesias (Sardinia), at the Palazzo Municipale, on 13th June 1923. Citizens of Iglesias! Black shirts! Fascisti! Your welcome, so cordial and so enthusiastic, surpasses any expectation. Iglesias has really been the cradle of Sardinian Fascismo. From here sprang the first groups of black shirts; it was, therefore, my definite duty to come and get into touch with you. You deserve that the Government should remember you, as in this island there is a large reserve of faith and ardent patriotism: I go back to Rome with my heart overcome with emotion. Since Italy has been united this is the first time that the head of the Government is in direct touch with the people of Sardinia. One thing only I regret, and that is that the shortness of my visit has not given me an opportunity of seeing more of your beautiful land. But I formally pledge myself to come again and visit your towns and your villages. As the head of the Government I am glad to have found myself amongst industrious, quiet and truly patient people, who have been too long forgotten, indeed almost considered as a far-away colony. It is well it should be known that Sardinia is one of the first regions of Italy, and it should be known, too, that she gave the largest contribution of lives to our glorious victory. As the head of the Government I am glad to find myself among the heroic black shirts and to have seen the splendid flourishing conditions of Fascismo, which will bring a complete regeneration to your land.
- 65. Here (said the Hon. Mussolini, putting his hand on the standard of Iglesias, which was hoisted near him)—here is the standard, the symbol of pure faith. I kiss it with fervour, and with the same fervour I embrace you, O magnificent people of Sardinia. (Loud applause.)
- 66. “AS WE HAVE REGAINED THE MASTERY OF THE AIR, WE DO NOT WANT THE SEA TO IMPRISON US” Speech delivered at Florence from the balcony of the Palazzo Vecchio, on 19th June 1923. Black shirts of Florence and Tuscany! Fascisti! People! Where shall I find the necessary words to express the fullness of my feelings at this moment? My words cannot but be inadequate for the purpose. Your solemn, enthusiastic welcome stirs me to the depths of my heart. But it is certain that it is not only to me that you pay this extraordinary honour, but also, I think, to the idea of which I have been the inflexible protagonist. Florence reminds me of the days when we were few. (Deafening applause.) Here we held the first glorious meeting of the Italian “Fasci di Combattimento.” You remember, we had often to interrupt our meeting to go out and drive away the base rabble. (“Bravo!” Frantic applause.) We were few then! Well, in spite of this huge crowd here assembled, I say that we are still few, not with regard to the enemies who have been put to flight for ever, but with regard to the enormous tasks that lie before our Italy. (Applause.) I said that our enemies have been put to flight, as we shall no more do the honour of considering as enemies certain corpses of the Italian political world—(“Bravo!”)—who delude themselves that they still exist simply because they abuse our generosity. Tell me, then, Black shirts of Tuscany and of Florence, were it necessary to begin again, should we begin again? (Deafening applause and cries of “Yes! Yes!”) This loud cry of yours, more than a promise, is an oath which seals for ever the Italy of the past, the Italy of the swindlers, of the
- 67. deceivers, of the pusillanimous, and opens the way to “our” Italy, the Italy whom we bear proudly in our hearts, who belongs to us who represent the new generation who adore strength, who is inspired by beauty, who is ready for anything when it is necessary to sacrifice herself to struggle and to die for the ideal. I tell you that Italy is going ahead. Two years ago, when the bestiality of the red demagogy raged, only twenty aeroplanes entered for the Baracca Cup. Last year they were thirty-five; this year, up to now, ninety. And as we have regained the mastery of the air, so we do not want the sea to imprison us. It must be, instead, the way for our necessary expansion in the world. (Great applause.) These, O Fascisti, Citizens, are the stupendous tasks which lie before us. And we shall not fail in our aim if each of you will engrave in his own heart the words by which is summed up the commandment of this ineffable hour of our history as a people: “Work,” which little by little must redeem us from foreign dependence; “Harmony,” which must make of the Italians one family; “Discipline,” by which at a given moment all Italians become one and march hand in hand towards the same goal. Black shirts! You feel that all the manœuvres of our adversaries tending to sever me from you are ridiculous and grotesque. And I hope it will not seem to you too proud a statement if I say that Fascismo, which I have guided on the consular roads of Rome, is solidly in our hand—(“Bravo!”)—and that if anybody should delude himself in this respect I should only need to make a sign, to give an order: “A noi!” (Deafening applause.) Raise up your standards! They have been consecrated by the sacred blood of our dead. When faith has thus been consecrated it cannot fail, cannot die, will not die! (Prolonged applause.)
- 68. “I PROMISE YOU—AND GOD IS MY WITNESS— THAT I SHALL CONTINUE NOW AND ALWAYS TO BE A HUMBLE SERVANT OF OUR ADORED ITALY” Speech delivered on 19th June 1923, at Florence, in the historical Salone dei Cinquecento, where the Municipal Council solemnly bestowed on Mussolini the freedom of the city of Florence. Mr. Mayor, Councillors, People of Florence, the capital for many centuries of Italian art,—You will notice that—on account of the honour which you pay me—I feel moved. To be made a citizen of Florence, of this city which has left such indelible traces on the history of humanity, represents a memorable and dominating event in my life. I do not know if I am really worthy of so much honour. (Cries of “Yes.” “May God preserve you for the future of our Italy.” Applause.) What I have done up to now is not much; but oh! Citizens of Florence, my determination is unshakable. (“Bravo!”) Human nature, which is always weak, may fail, but not my spirit, which is dominated by a moral and material faith—the faith of the country. From the moment in which Italian Fascismo raised its standards, lit its torches, cauterised the sores which infected the body of our divine country, we Italians, who felt proud to be Italians—(“Bravo! Bravo!” Applause.)—are in spiritual communion through this new faith. Citizens of Florence! I make you a promise, and be sure I shall keep it! I promise you—and God is my witness in this moment of the purity of my faith—I promise you that I shall continue now and
- 69. always to be a humble servant of our adored Italy! (Prolonged applause.)
- 70. “THE VICTORY OF THE PIAVE WAS THE DECIDING FACTOR OF THE WAR” Speech delivered in Rome on 25th June 1923, from Palazzo Venezia, in commemoration of the anniversary of the Battle of the Piave. Fellow-Soldiers!—After your ranks, so well disciplined and of such fine bearing, have marched past His Majesty the King, the intangible symbol of the country, after the austere ceremony in its silent solemnity before the tomb of the Unknown Warrior, after this formidable display of sacred strength, words from me are absolutely superfluous, and I do not intend to make a speech. The march of to- day is a manifestation full of significance and warning. A whole people in arms has met to-day in spirit in the Eternal City. It is a whole people who, above unavoidable party differences, finds itself strongly united when the safety of the common Motherland is at stake. On the occasion of the Etna eruption, national solidarity was wonderfully manifested; from every town, every village, one might say from every hamlet, a fraternal heart-throb went out to the land stricken by calamity. To-day tens of thousands of soldiers, thousands of standards, with men coming to Rome from all parts of Italy and from the far-away Colonies, from abroad, bear witness that the unity of the Italian nation is an accomplished and irrevocable fact. After seven months of Government, to talk to you, my comrades of the trenches, is the highest honour which could fall to my lot. And I do not say this in order to flatter you, nor to pay you a tribute which might seem formal on an occasion like this. I have the right to interpret the thoughts of this meeting, which gathers to listen to my
- 71. words as an expression of solidarity with the national Government. (Cries of assent.) Let us not utter useless and fantastical words. Nobody attacks the sacred liberty of the Italian people. But I ask you: Should there be liberty to maim victory? (Cries of “No! no!”) Should there be liberty to strike at the nation? Should there be liberty for those who have as their programme the overthrow of our national institutions? (Cries of “No! no!”) I repeat what I explicitly said before. I do not feel myself infallible, I feel myself a man like you. I do not repulse, I cannot, I shall not repulse any loyal and sincere collaboration. Fellow-soldiers! The task which weighs on my shoulders, but also on yours, is simply immense, and to it we shall be pledged for many years. It is, therefore, necessary not to waste, but to treasure and utilise all the energies which could be turned to the good of our country. Five years have passed since the battle of the Piave, from that victory on which it is impossible to sophisticate either within or beyond the frontier. It is necessary to proclaim, for you who listen to me, and also for those who read what I say, that the victory of the Piave was the deciding factor of the war.... On the Piave the Austro- Hungarian Empire went to pieces, from the Piave started its flight on white wings the victory of the people in arms. The Government means to exalt the spiritual strength which rises out of the victory of a people in arms. It does not mean to disperse them, because it represents the sacred seed of the future. The more distant we get from those days, from that memorable victory, the more they seem to us wonderful, the more the victory appears enveloped in a halo of legend. In such a victory everybody would wish to have taken part! We must win the Peace! Too late somebody perceived that when the country is in danger the duty of all citizens, from the highest to the lowest, is only one: to fight, to suffer and, if needs be, to die! We have won the war, we have demolished an Empire which threatened our frontiers, stifled us and held us for ever under the
- 72. extortion of armed menace. History has no end. Comrades! The history of peoples is not measured by years, but by tens of years, by centuries. This manifestation of yours is an infallible sign of the vitality of the Italian people. The phrase “we must win the peace” is not an empty one. It contains a profound truth. Peace is won by harmony, by work and by discipline. This is the new gospel which has been opened before the eyes of the new generations who have come out of the trenches; a gospel simple and straightforward, which takes into account all the elements, which utilises all the energies, which does not lend itself to tyrannies of grotesque exclusivism, because it has one sole aim, a common aim: the greatness and the salvation of the nation! Fellow-soldiers! You have come to Rome, and it is natural, I dare to say, fated! Because Rome is always, as it will be to-morrow and in the centuries to come, the living heart of our race! It is the imperishable symbol of our vitality as a people. Who holds Rome, holds the nation! The “Black Shirts” buried the Past. I assure you, my fellow- soldiers, that my Government, in spite of the manifest or hidden difficulties, will keep its pledges. It is the Government of Vittorio Veneto. You feel it and you know it. And if you did not believe it, you would not be here assembled in this square. Carry back to your towns, to your lands, to your houses, distant but near to my heart, the vigorous impression of this meeting. Keep the flame burning, because that which has not been, may be, because if victory was maimed once, it does not follow that it can be maimed a second time! (Loud cheers, repeated cries of “We swear it!”) I keep in mind your oath. I count upon you as I count upon all good Italians, but I count, above all, upon you, because you are of my generation, because you have come out from the bloody filth of the trenches, because you have lived and struggled and suffered in the face of death, because you have fulfilled your duty and have the
- 73. right to vindicate that to which you are entitled, not only from the material but from the moral point of view. I tell you, I swear to you, that the time is passed for ever when fighters returning from the trenches had to be ashamed of themselves, the time when, owing to the threatening attitudes of Communists, the officers received the cowardly advice to dress in plain clothes. (Applause.) All that is buried. You must not forget, and nobody forgets, that seven months ago fifty-two thousand armed “black shirts” came to Rome to bury the past! (Loud cheers.) Soldiers! Fellow-Soldiers! Let us raise before our great unknown comrade the cry, which sums up our faith: Long live the King! Long live Italy, victorious, impregnable, immortal! (Loud cheers, whilst all the flags are raised and waved amidst the enthusiasm of the immense crowd in the square.)
- 74. THE RELATIONS BETWEEN ITALY AND THE UNITED STATES Speech by the American Ambassador to Rome. On the 28th June 1923 the Italo-American Association held in Rome a banquet in honour of Mr. Richard Washburn Child, American Ambassador to Italy, and of the Hon. Mussolini, President of the Italian Council. The two distinguished guests delivered the following speeches,[14] which have a special importance, both with regard to Fascismo and to Italo-American relations. 14. The two speeches have been courteously given at his request to Baron Quaranta di San Severino for publication by the American Ambassador, Richard Washburn Child. The object of this meeting was clearly explained by the Hon. Baron Sardi, Italian Under-Secretary of State for Public Works, in an appropriate address to the illustrious guests (published in full by the Bulletin of the Library for American Studies in Italy, No. 5), in which, after having thanked them in the name of Senator Ruffini, President of the Association, still detained on account of important duties in Geneva, and also in the name of the other members, for the honour they conferred on the Society by their presence, went on to lay stress on the purpose for which the Association exists, namely, to promote a better reciprocal understanding between the American and Italian peoples through the manifold activities of their respective countries. The Hon. Sardi announced that during the summer months of this year courses of preparation will be inaugurated again for American students who wish to come and visit our country and study our language, literature and history, while for next October, under the patronage of the American Ambassador and the Italian Premier, with the co-operation of American and Italian professors, special industrial and commercial courses are in preparation. The American students will be able to benefit by the use of the valuable library of the Association, which is daily enriched by the competent work of Commendatore Harry Nelson Gay and his collaborators. The Hon. Sardi, after referring to the fraternity of arms, which during the Great War brought together the soldiers of Italy and America, said that, having returned now to the peaceful spheres of industry and culture, these forms of effort
- 75. contribute strongly to cement between the two countries that spiritual fraternity which arises out of a better mutual acquaintance with the respective virtues and qualities and a clearer realisation of our aspirations. The orator concluded by expressing the wish that the Italo-American Association, by the indissoluble union of cultured minds, might be able to intensify the bonds already uniting the United States of America and Italy. Mr. President and Gentlemen,—It is my privilege to propose a toast to the King and to the spirit of an Italy now stronger and more united than ever before. I wish to express the earnest hope that my country and yours will continue to stand together in upholding ideals which make men strong instead of tolerating those which make men weak. During the last eight months Italy has made an extraordinary contribution to the whole world by raising ideals of human courage, discipline, and responsibility. I would be unfaithful to my beliefs and to those of hosts of Americans if I failed to acknowledge the part played by your President of Council, Mussolini, with the people of Italy, in giving to all mankind an example of courageous national organisation founded upon the disciplined responsibility of the individual to the State, upon the abandonment of false hopes in feeble doctrines, and upon appeal to the full vigorous strength of the human spirit. We have heard a great deal in the last few years about the menace which war brings before the face of the world. I am confident that my people and your people are willing to act together to contribute anything possible to reduce the dangers of war, but I hold the belief, and I think your Premier holds the belief, that worse menaces than war now oppose the progress of mankind. Folly and weakness and decay are worse. These menaces of weakness are often fostered by men of good intentions, who talk about the need to rescue mankind and about the necessity to establish the rights of mankind. I want to see leaders of men who, instead of teaching humanity to look outside themselves for help, will teach humanity that it has
- 76. power within itself to relieve its own distress. I want to see leaders who, instead of telling men of their rights, will lead them to take a full share of their responsibilities. I do not doubt that the spirit of benevolence is a precious possession of mankind, but a more precious possession is the spirit which raises the strength of humanity so that benevolence itself becomes less of a necessity. He who makes himself strong and calls upon others to be strong is even more kind and loving of the world than he who encourages men to seek dependence on forces outside themselves or upon impracticable plans for new social structures. I do not doubt the good faith of many of those who put forth theories of new arrangements of social, economic and international structure, but they may all be sure that more important than any of these theories is individual responsibility and the growth and spread of self-reliance in the home and in the nation. I do not doubt that we, Italians and Americans, have a full appreciation of the pity which we ought to confer upon weak or wailing groups or nations or races which clamour for help or favour; but I trust that, even in the competition of peace or war, I shall be the last ever to believe that weak groups or nations or races are superior or are more worthy of my affection than those who mind their own business with industry, strength and courage, and stand upon their own strong legs. I do not question the motives of many of those who, feeling affectionate regard for the welfare of their fellow-men, hope for a structure of society in which international bodies shall hand down benefactions to communities, and communities shall hand down benefactions to individuals. I merely point out that some nations, such as yours and mine, are beginning to believe that these ideas come out of thoughts which, though easily adopted, are the offspring of a marriage of benevolence with ignorance. In any structure of society which can command our respect and our faith the current of responsibility runs the other way. The doctrine that the world’s strength arises from the responsibility of the individual is
- 77. a sterner doctrine. The leaders of men who insist upon it are those who will be owed an eternal debt by mankind. The strength of society must come from the bottom upward. The world needs now more than anything else the doctrine that the first place to develop strength is at home, the first duty is the nearest duty. A strong co-operation of nations can only be made of nations which are strong nations, a strong nation can only be made of good and strong individuals. When one makes the fasces, the first requirement is to find the individual rods, straight, strong and wiry, such as you have found, Mr. President, and so skilfully bound together in the strength of unity. But if they had been rotten sticks you could not have made the fasces. Unity in action would have been impossible. The rotten sticks would have fallen to pieces in your fingers. Mr. President, what the world needs is not better theories and dreams, but better men to carry them out. The world needs a spirit which thinks first of responsibilities before it thinks of rights. It was this spirit which you have done so much to awaken into new life in Italy. Not long ago I heard a speech made by a foreigner in Italy who is used to dealing with economic statistics. He was trying to account for the new life in Italy on the basis of comparative statistics. I told him he could not do it until he could produce statistics of the human spirit. I told him he could not account for everything in Italy until he could reduce to statistics that wonderful record of the human spirit which in scarcely more than half a century has created the new Italy. I told him he would have to account for the number of Italians who in 1848 and 1859, in the Great War and 1923, had a cause for which they were willing to die. I told him that I was always a nationalist before I was an internationalist, and I would go on being a nationalist, believing in the spirit of strong and upright and generous nationalism, and believing not in theorising nations or whining peoples, but in nations and peoples who develop a national spirit so finely tempered that they offer to the world an example of organisation, discipline and fair play, because they themselves are
- 78. upright and strong men and can contribute valuably to international co-operation. I said to him that when he could produce statistics on human virtues and human spirit he would be nearer to understanding what made progress in the world. I asked him if he had figures to show the difference between nations which breed men who are ready to die for their beliefs and nations which produce no such men. I asked him to put his figures back in his pocket and go out and talk to the youth of Italy. Mr. President, the youth of Italy, as in any other country, are the trustees of the spirit of to-morrow. It is a fact which goes almost unnoticed, that the training of masses of youth in the spirit of discipline and fair competition and of loyalty to a cause is largely to be found in athletic games. It is a fact which almost always is forgotten, that nations of history or those of to-day which have engaged in athletic games are the strong nations, and those which have had no athletics are the weak nations. It is a fact almost neglected that nations which can express their spirit of competition in athletics are the nations which have the least destructive restlessness within and are the most fair and, indeed, are the most restrained in their dealings with other nations. Athletic games teach the lesson that every man who competes must win by reason of his own virtue. No help can come from without. There is no special privilege for anyone. He who wins does so by merit alone. Athletic games, whenever they are carried on by teams, teach the lesson that the individual must put aside his own interests for the good of his group. There must be a voluntary submission to discipline and absolute loyalty to a captain in order to avoid the humiliation of disorganisation and defeat. Athletic games are not for the weak and complaining, but for the strong and for the lovers of fair play. Finally, they furnish oft-repeated lessons of the truth that when flesh and muscles and material agencies seem about to fail, human will and human spirit can work miracles of victory.
- 79. Because I believe in these ideals for my own country and for yours, I offer through you, for the purposes which the Olympic Committee of Italy will set forth, a small but friendly token of my deep interest in the youth of Italy. (Loud applause.) The Italian Prime Minister’s Reply Mr. Ambassador,—The discourse which your Excellency has pronounced at this reunion strengthens the bonds of sympathy and fraternity between Italy and America, and has profoundly interested me in my capacity as an Italian and as a Fascista. As an Italian, because you have spoken frank words of cordial approval of the Government which I have the honour to direct. I have no need to add that this cordiality is reciprocated by me and by all Italians. There is no doubt that the elements for a practical collaboration between the two countries exist. It is only a question of organising this collaboration. Some things have been done, but more remain to be done. I will not surprise your Excellency if I point out, without going into particulars, a problem which concerns us directly. I speak of the problem of emigration. I limit myself only to saying that Italy would greet with satisfaction an opening in the somewhat rigid meshes of the Immigration Bill, so that there could be an increase in Italian emigration to North America, and would greet with similar satisfaction the employment of American capital in Italian enterprises. As a Fascista, the words of your Excellency have interested me because they reveal an exact understanding of the phenomenon and of our movement, and constitute a sympathetic and powerful vindication of it. This fact is the more remarkable because the Fascismo movement is so complex that the mind of a stranger is not always the best adapted to understand it. You, Mr. Ambassador, constitute the most brilliant exception to this rule. Your discourse, I say, contains all the philosophy of Fascismo and of the Fascismo endeavour, interwoven with an exaltation of strength, of beauty, of discipline, of authority, and of the sense of responsibility. You have been able to show, Mr. Ambassador, that in spite of the numerous difficulties of the general situation, Fascismo has kept
- 80. faith to its promises given before the “March on Rome.” The time intervening since those promises were made has been short, so that only a stupid person would pretend that the work is already completed. I limit myself to saying that I find corroboration by your Excellency that it is well begun. I am certain, Mr. Ambassador, that all Italians will read with emotion the words which you have pronounced on this memorable occasion. I ask you especially to believe this. I have heard, just now, not a discourse in the manner and strain of an ordinary conventional speech, but a clear and inspiring exposition of the conception of life and history which animates Italian Fascismo. I do not believe that I exaggerate when I say that this conception finds strong and numerous partisans even on the other side of the ocean, among the citizens of a people who have not the thousands of years of history behind them which we have, but who march to-day in the vanguard of human progress. In this affinity of conceptions I find the solid basis for the fraternal understanding between Italy and America. The announcement that you, Mr. Ambassador, are giving a wreath of gold to the Italian youth who will be victor in the next Olympic competition games will win the hearts of all Italian athletes, and of these there are, as you know, innumerable legions. I thank your Excellency in the name of Italian youth, almost all of whom have put on the “black shirt,” especially the young athletes, and, at the same time that I encourage the Italo-American Society to persevere in the execution of its splendid programme, I declare that my Government will do whatever is necessary to develop and strengthen the economic and political relations between the United States and Italy. I raise my glass to the health of President Harding and the fortunes of the great American people. (Loud applause.)
- 81. “THE GREATNESS OF THE COUNTRY WILL BE ACHIEVED BY THE NEW GENERATIONS” Speech delivered 2nd July 1923 in Rome, at the Palazzo Venezia, before the schoolboys of Trieste, Nicastro, Castelgandolfo, Vetralla and Perugia and their masters, who were accompanied by representatives of the Roman “balillas,” and had come to Rome to pay homage at the tomb of the “Unknown Warrior,” before which they laid a wreath of beaten iron and kneeling repeated the oath of love and loyalty to the King and the Country. The Hon. Mussolini with the Minister of War, General Diaz; the Under-Secretary of State for the Presidency, Hon. Acerbo; General De Bono, the Director General of Police; Signor Lombardo Radice, the Director General of Primary Schools, and other officials, greeted them. The Hon. Mussolini thus addressed the meeting: On this radiant morning you have offered the capital a magnificent spectacle. Romans, having lived through many millenniums of history, are rather slow in being impressed by events and are not easily to be carried away by excessive enthusiasm. They have certainly however been filled to-day with admiration at this scene of promising youth which has been offered them by the schoolboys here gathered from all parts of Italy and especially from the “Venezia Giulia,” particularly dear to the heart of all Italians. It was well said that in the dark pre-war days the schools of the National League and in general the schools entrusted to Italian masters represented the centre around which were nursed the hopes and the faith of the Italian race. I am glad to express to you the feelings of my brotherly sympathy. I am pleased to add that the National Government, the Fascista Government, holds in high esteem the scholarly characteristics and has deep respect for the teachers of all grades, of all schools. The Fascista Government feels and knows that the greatness of the country, to which all of us must consecrate the best of our
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