Automorphisms of strongly regular graphs with applications to partial difference sets

Abstract

In this article we generalize a theorem of Benson (J Algebra 15:443-454, 1970) for generalized quadrangles to strongly regular graphs, deriving numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then use this result to develop a few new techniques to study regular partial difference sets (PDS) in Abelian groups. Ma (Des Codes Cryptogr 4:221-261, 1994) provided a list of parameter sets of regular PDS with k ≤ 100 in Abelian groups for which existence was known or had not been excluded. In particular there were 32 parameter sets for which existence was not known. Ma (J Stat Plan Inference 62:47-56, 1997) excluded 13 of these parameter sets. As an application of our results we here exclude the existence of a regular partial difference set for all but two of the undecided parameter sets from Ma's list.

New necessary conditions on Paley‐type partial difference sets in abelian groups

“…Below we cite three results on abelian regular PDSs. The first of these was proved by Ma [7], the second one was proved by Arasu et al [1], and the last one, the local multiplier theorem, was proved by De Winter, Kamischke, and Wang [3].…”

“…Then, up to complementation, Γ is either of Paley type or it has parameters (243, 22, 1, 2). Theorem 2.3 (De Winter, Kamischke, and Wang [3]). Let D be a regular v k λ μ ( , , , )-PDS in an abelian group G. Furthermore, assume Δ is a perfect square.…”

Restrictions on parameters of partial difference sets in nonabelian groups

“…For further applications of partial difference sets to coding theory and finite geometry, see the survey of Ma [17]. The case when G is abelian has been thoroughly studied; see [17] for a survey of older results and [5,6,7,8,11,18,19,20,23,24] for a number of very recent results. On the other hand, comparatively little is known in the case when G is nonabelian.…”

Non-existence of two types of partial difference sets

“…In this note we will prove nonexistence of such PDS, hence finalizing the classification of parameters for which a PDS with k ≤ 100 exists in an Abelian group. The proof uses ideas developed in [1], but requires an additional argument based on weighing points and lines in a projective plane.…”