Abelian Projective Planes of Square Order

We prove that, under certain conditions, multipliers of an abelian projective plane of square order have odd order modulo v*, where v* is the exponent of the underlying Singer group. As a consequence, we are able to establish the non-existence of an infinite number of abelian projective planes of square order

On Wilbrink's Theorem

AbstractAn easy extension of Wilbrink's Theorem on planar difference sets for higher values of λ is given. It follows that the Bruck-Ryser-Hall conjecture on the super-fluousness of the “p > λ” condition (or its variations thereof) in the multiplier theorems implies Hall's conjecture on the existence of only a finite number of (ν, κ, λ) abelian difference sets, for the case λ odd and κ − λ ≡ 2 (mod 4). More strongly, under these conditions we can show that the corresponding designs are Hadamard, i.e., k = 2λ + 1 and ν = 2k + 1

Matrix constructions of divisible designs

AbstractWe present two new constructions of group divisible designs. We use skew-symmetric Hadamard matrices and certain strongly regular graphs together with (v, k, λ)-designs. We include many examples, in particular several new series of divisible difference sets

A two-to-one map and abelian affine difference sets

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set H^^~ = H∖{1,ω}, where H = G/N and ω=∏_σ. Using D we define a two-to-one map g from H^^~ to N. The map g satisfies g(σ^m) = g(σ)^m and g(σ) = g(σ^) for any multiplier m of D and any element σ∈H^^~. As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N

Some New Results on Circulant Weighing Matrices

10.1023/A:1011903510338Journal of Algebraic Combinatorics14291-101JAOM

Study of proper circulant weighing matrices with weight 9

10.1016/j.disc.2004.12.029Discrete Mathematics308132802-2809DSMH

Abelian Difference Sets Without Self-conjugacy

Designs, Codes, and Cryptography153223-230DCCR