Abstract

We investigate symmetric divisible designs with parameters (m, n, k, λ₁, λ₂) with k − λ₁ = 1. We characterize such designs by their intersection numbers and give a construction method using strongly regular graphs with λ = μ − 1; we thus obtain a new infinite family of examples. We then consider the special case of symmetric divisible designs with parameters (m, n, k, λ₁, λ₂) admitting an abelian Singer group; equivalently, we study abelian divisible difference sets with parameters (m, n, k, k − 1, λ₂). Improving results of a previous paper, we show that such a DDS is either reversible (i.e., fixed under inversion) and arises from a partial difference set with parameter β = -1 (i.e., a strongly regular graph with λ = μ − 1 admitting a Singer group) or arises (by a new construction) from a Paley Hadamard difference set. In the second case, all possible parameters have been determined. In the first case, the DDS is known to be equivalent to a partial difference set with β = -1 (as we have shown in [1]); using this, certain restrictions were obtained in that paper. We will here give two further restrictions, but a complete classification of the possible parameters is as yet missing. However, we can obtain a complete classification of all cyclic divisible difference sets satisfying f − λ₁ = 1.

Article Outline

• References

^* Research partially supported by NSA grant # MDA 904-87-H-2018 and by an Alexander-von-Humboldt fellowship. The author would like to thank the Mathematisches Institut der Justus-Liebig-Universität Gieβen for its hospitality during the time of this research.