Judith Q. Longyear

Classification of 3-(24, 12, 5) designs and 24-dimensional Hadamard matrices

Recently the authors completed the classification of 3-(24, 12, 5) designs up to isomorphism. These designs are closely related to 24-dimensional Hadamard matrices, and the work on designs leads to a classification of the matrices up to equivalence. Hadamard matrices of lower dimension had been determined previously by Hall [2-41. This article described the techniques used in the classification and contains a summary of the results. It provides information from which each of the 129 designs and 59 matrices can be constructed, and it gives the order and orbit lengths of the automorphism group of each design and matrix. An explicit list of the matrices, together with generating permutations for their automorphism groups, may be found in [5]. The corresponding data for designs appear in 161. In general, a 3-(413. + 4, 21 + 2,L) design D = (P, B) is called a Hadamard design (P and B denote the sets of points and blocks, respectively). In such a design the set of 8A + 6 blocks is necessarily closed under complementation [7]; thus, if 6 denotes P a, B consists of 4,l + 3 block pairs 01* = {a, C}, which we term parallel classes. For a E P, we let B, denote the set of blocks containing the point a.

Regular d-valent graphs of girth 6 and 2(d2−d+1) vertices

Abstract For each d such that d -1 is prime, a d -valent graph of girth 6 having 2( d 2 − d +1) vertices is exhibited. The method also gives the trivalent graph of girth 8 and 30 vertices.

Nested group divisible designs and small nested designs

Abstract The designs considered are nested in the sense of W.T. Federer, that is, each block of the larger design contains a smaller distinguished subblock and the set of subblocks also forms a design. One of the most useful constructions for obtaining new designs is the group divisible design. We use the concept of nested group divisible design to obtain many infinite classes of nested group divisible designs and nested designs and give a catalogue of all possible nested designs with larger replication number ≤15.

Arrays of strength s on two symbols

Abstract This paper gives necessary and sufficient conditions on σ , s , t and on μ , s , t for an array with s + t rows to have strength s and weight σ , or to be balanced and have strength s and weight μ . If a balanced array can exist, the conditions provide a construction. The solutions for t =1,2 are also given in an alternate form useful for the study of trim arrays. The balanced solution for t =1 is more detailed than that known so far, and permits one to determine whether or not a solution exists in possibly fewer steps.

A peculiar partition formula

Abstract Let p = (a1,…,an,d) be a partition of n=a1(1)+a2(2)+⋯+an(n) with d=n+2-(a1+⋯+an) then 3n+3 n = ∑ p n+2 a 1 ,…,a n ,d prod; i=1 n (i+1) a i The proof of this formula involves using graphs to count the number of ways in which a permutation can reasonably be written as the product of transpositions.

Order 24 Hadamard matrices of character at least 3

Abstract It is determined that the number of inequivalent Hadamard matrices of order 24 and character at least 3 is at least 29. N. Ito, J. Leon and the author are determining the equivalence classes among the 133 matrices given here. The author has shown elsewhere [in “Theory and Applications of Graphs,” Lecture Notes in Mathematics No. 642, Springer-Verlag, Berlin/Heidelberg/New York, 1978; and in “Proceedings of the April 1978 Meeting of the New York Academy of Sciences”] that all Hadamard matrices of character 2 and order 24 have higher transpose character and that there is exactly one matrix of order 24 with both characters one.

THERE IS ONE HADAMARD MATRIX OF ORDER 24 AND BOTH CHARACTERS 1

This paper is a part of the joint effort of N. Ito, J. Leon, and the author to determine the number of Hadamard matrices of order 24 which are inequivalent under transposition, row, or column permutation and row or column negation. The matrices of character at least 2 in either the plain or transpose have been discussed earlier. Here we show there is a unique matrix with both characters one.

Some isomorphisms between pairs of latin squares

Let A, B, C, D be latin squares with A orthogonal to B and C orthogonal to D. The pair A, B is isomorphic with the pair C, D if the graph of A, B is graph-isomorphic with the graph of C, D. A characterization is given for determining when a pair A, B of latin squares is isomorphic with a self-orthogonal square C and its transpose. Self-orthogonal squares are important because they are both abundant and easy to store. An algorithm either displays a self-orthogonal square C and an isomorphism from A, B to C, CT or, if none exists, gives a small set of blocks to the existence of such a square isomorphism.

Common transversals in partitioning families

A family of s subsets of a finite set R is a partitioning family of R if the subsets are pairwise disjoint. A transversal of such a family is a set of s elements, one in each member of the family. If there are several partitioning families defined on R, each having s members, then a set X⊆R is a common transversal if X is a transversal for each family. n nReducing R by inclusion equivalence provides a uniform cardinality condition which is sufficient to insure that the families have a common transversal. This bound is best possible, since any smaller cardinality permits some subset to be void. If it is known that no subset is empty, a much smaller bound can be obtained for t=2, using Halls condition. A further application gives a bound for t families guaranteeing that each pair has a common transversal. n nLet b(i:s,t,k) be the least number such that in t partitioning families of s sets each, every i of the families have a common transversal when it is known that every set has at least k elements. nThen it is shown that: n1. n(i) nb(t:s,t,0)=(s−1)s1−1+1 n, n n2. n(ii) nb(2:s,2,1)=(s−2)s+3 nfor s⩾3, n n3. n(iii) nb(2:s,t,0)=b(t:s,t,0) nfor s⩾3, n n4. n(iv) nb(2:s,t,1)=s1−2(s2-2s+2)+1 nfor s⩾3, n n5. n(v) nb(t:s,t,1)>(s−2)st−1+st−2+(t−2)(s−1)t−2+1 nfor s⩾3.

Graphs and Permutations

DEFINITION 2: A set of differences A = {al, . . . , u R ] (mod N ) is a set of R residues in the ring of integers modulo N . This is not to be confused with a perfect difference set that has the further requirement that every nonzero residue A must occur the same number of times as some ui aj = A (mod N ) . If B = { b l , . . . , b R } is a set of differences and there is some residue Wand some permutation A such that bi + W = a,(i) (mod N ) , then B is called a displacement of A.