K. T. Arasu

The solution of the Waterloo problem

Abstract Let D(d, q) be a classical (ν, k, λ)-Singer difference set in a cyclic group G corresponding to the complement of the point-hyperplane design of PG(d, q) (d ⩾ 1). We characterize those Singer difference sets D(d, q) which admit a “Waterloo decomposition” D = A ∪ B such that (A − B) · (A − B)(−1) = k in Z G: Theorem. D(d, q) admits a Waterloo decomposition if and only if d is even.

Divisible difference sets with multiplier −1

Abstract We investigate proper ( m , n , k , λ 1 , λ 2 )-divisible difference sets D in an abelian group G admitting the multiplier − 1. We show that this assumption implies severe restriction on the parameters of D and the structure of G . For instance, if D is even reversible (i.e., D is fixed by the multiplier − 1), the square-free part of k − λ 1 has to be 1 or 2. In the case of relative difference sets (i.e., in the case λ 1 = 0), one necessarily has k = m = nλ 2 , and thus the associated symmetric divisible design dev D has to be a symmetric transversal design. We also construct some new series of examples, among them an infinite series of relative difference sets D with “weak” multiplier − 1 (i.e., − 1 fixes no translate of D but still induces an automorphism of dev D —a situation which cannot arise for ordinary difference sets). Finally, we partially characterize the (reversible) divisible difference sets with k − λ 1 ⩽1; moreover, we obtain a complete characterization of all cyclic reversible divisible difference sets for which n is even.

New constructions of Menon difference sets

Abstract Menon difference sets have parameters (4N2, 2N2 − N, N2 − N). These have been constructed for N = 2a3b, 0 ⩽ a,b, but the only known constructions in abelian groups require that the Sylow 3-subgroup be elementary abelian (there are some nonabelian examples). This paper provides a construction of difference sets in higher exponent groups, and this provides new examples of perfect binary arrays.

On Abelian Difference Sets

We review some existence and nonexistence results — new and old — on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics.

Affine difference sets of even order

Abstract Generalizing a result of Ko and Ray-Chaudhuri (Discrete Math. 39 (1982), 37–58), we show the following: Assume the existence of an affine difference set in G relative to N of even order n≠2. If G is of the form G = N ⊕ H, where N is abelian, then n is actually a multiple of 4, say n = 4k, and there exists a (4k − 1, 2k − 1, k − 1)-Hadamard difference set in N. More detailed considerations lead to variations of this result (under appropriate assumptions) which yield even stronger non-existence theorems. In particular, we show the non-existence of abelian affine difference sets of order n ≡ 4 mod 8 (with the exception n = 4) and of nilpotent affine difference sets of order n ≡ 2 mod 4 (n ≠ 2). The latter result is the first general non-existence theorem in the non-abelian case.

A nonexistence result for abelian menon difference sets using perfect binary arrays

AbstractA Menon difference set has the parameters (4N2,2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian groupn

Matrix constructions of divisible designs

H times K times Z_{p^alpha }

The mann test for divisible difference sets

n contains a Menon difference set, wherep is an odd prime, |K|=pα, andpj≡−1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian groupn