G. H. J. van Rees

An Enumeration of Binary Self-Dual Codes of Length 32

A binary self-dual code of length 2k is a (2k, k) binary linear code C with the property that every pair of codewords in C are orthogonal. Two self-dual codes, C1 and C2, are equivalent if and only if there is a permutation of the coordinates of C1 that takes C1 into C2. The automorphism group of a binary code C is the set of all permutations of the coordinates of C that takes C into itself.The main topic of this paper is the enumeration of inequivalent binary self-dual codes. We have developed algorithms that will take lists of inequivalent small codes and produce lists of larger codes where each inequivalent code occurs only a few times. We have defined a canonical form for codes that allowed us to eliminate the overenumeration. So we have lists of inequivalent binary self-dual codes of length up to 32. The enumeration of the length 32 codes is new. Our algorithm also finds the size of the automorphism group so that we can compute the number of distinct binary self-dual codes for a specific length. This number can also be found by counting and matches our total.

The equivalence of certain equidistant binary codes and symmetric bibds

We study equidistant codes of length 4k + 1 having (constant) weight 2k, and (constant) distance 2k between codewords. The maximum number of codewords is 4k; this can be attained if and only ifk = (u2 +u)/2 (for some integeru) and there exists a ((2u2 + 2u + 1,u2, (u2 −u)/2) — SBIBD. Also, one can construct such a code, with 4k − 1 codewords, from a (4k − 1, 2k − 1,k − 1) — SBIBD.

V(m,t)'s for m=4,5,6☆

The spectrum for V(m,t) is solved for m=4,5 and 6 using ad hoc methods for m= 4 and using exponential sums for m=5 and 6. A V(m,t) leads to m idempotent pairwise orthogonal Latin squares of order (m+1)t+1 with one common hole of order t.

Maximal sets of mutually orthogonal Latin squares

Maximal sets of s mutually orthogonal Latin squares of order v are constructed for innitely many new pairs (s;v). c 1999 Published by Elsevier Science B.V. All rights reserved

V(m,t) and its variants☆

Abstract A V ( m , t ) leads to m idempotent pairwise orthogonal Latin squares of order ( m +1) t +1 with one common hole of order t . For m =3,4,5 and 6 the spectrum for V ( m , t ) has been determined recently by Ling et al. In this article, Weils theorem on character sums is used to get the spectra for V ( m , t )s, for m =7. For variant V ( m , t )s, such as V (2) ( m , t ) with m =2,4,6 and V (4) ( m , t ) with m =2,4, the spectrums are also determined. Three infinite families of V λ ( m , t )s with λ=2, m=2; λ=2, m=3 and λ=3, m=2 are proved to exist.

Splitting systems and separating systems

Abstract Suppose m and t are integers such that 0 (X, B ) where |X|=m, B is a set of ⌊m/2⌋ subsets of X, called blocks such that for every Y⊆X and |Y|=t, there exists a block B∈ B such that |B∩Y|=⌊t/2⌋ or |(X⧹B)∩Y|=⌊t/2⌋. We will give some results on splitting systems for t=2 or 4 which often depend on results from uniform separating systems. Suppose that m is an even integer, t 1 , t 2 are integers such that t1+t2⩽m. A uniform (m,t1,t2)-separating system is an ordered pair (X, B ) where |X|=m, B is a set of subsets of X of size m/2, called blocks, such that for every P⊆X, Q⊆X where |P|=t 1 , |Q|=t 2 and P∩Q=∅, there exists a block B∈ B for which either P⊆B, Q∩B=∅ or Q⊆B, P∩B=∅ . We also give new results for separating systems.

(22, 33, 12, 8, 4)-BIBD, an Update

A (v, b,r, k, λ)-balanced incomplete block design is a family of b sets (called blocks) of size k whose elements (varieties) are from a v-set, v > k, such that every element occurs exactly r times and every pair exactly λ times. A (22, 33, 12, 8, 4)-BIBD is the set of parameters with the smallest v for which it is not known whether a BIBD exists or not. A survey of what is known about such a design is given.

Constructions and bounds for (m,t)-splitting systems

Let m and t be positive integers with t>=2. An (m,t)-splitting system is a pair (X,B) where |X|=m and B is a collection of subsets of X called blocks such that for every Y@?X with |Y|=t, there exists a block B@?B such that |B@?Y|=@?t/2@?. An (m,t)-splitting system is uniform if every block has size @?m/2@?. In this paper, we give several constructions and bounds for splitting systems, concentrating mainly on the case t=3. We consider uniform splitting systems as well as other splitting systems with special properties, including disjunct and regular splitting systems. Some of these systems have interesting connections with other types of set systems.

A new family of BIBD's and non-embeddable (16, 24, 9, 6, 3)-designs

Abstract We construct a new family of balanced incomplete block designs with parameters (2n 2 +3n+2, ( (n+1) 2 )(2n 2 +3n+2), (n+2) 2 , 2n+2, n+1) where n and n +1 are prime powers. Also we construct 251 non-embeddable (16, 24, 9, 6, 3) designs and thereby increasing the lower bound on the number of pairwise non-isomorphic balanced incomplete block designs (16, 24, 9, 6, 3) to 1542.

Many (22, 44, 14, 7, 4) - and (15, 42, 14, 5, 4)-balanced incomplete block designs

Abstract A Balanced Incomplete Block Design (BIBD) is a pair (V, B) where V is a v-set and B is a collection of b k-subsets of V, called blocks, such that every element of V occurs in exactly r of the k-subsets and every 2-subset of V occurs in exactly λ of the blocks. The number of non-isomorphic designs of a BIBD (22, 44, 14, 7, 4) whose automorphism group is divisible by 7 or 11 are investigated. From this work, results are obtained on the number of non-isomorphic BIBDs (15, 42, 14, 5, 4).