Strongly Regular Graphs with Generalized Denniston and Dual Generalized Denniston Parameters
Shuxing Li, James A. Davis, Sophie Huczynska, Laura Johnson, John Polhill
TL;DR
The paper addresses constructing SRGs and PDSs with generalized Denniston and dual generalized Denniston parameters in elementary abelian groups. It introduces two infinite families based on unions of cyclotomic classes indexed by subspaces, proves these subsets are PDSs and thus yield SRGs with explicit parameter formulas, and develops dual constructions linking to dual Denniston parameters. By detailing how these families recover and unify a range of prior results (Denniston, Momihara, Ott, BXZ, DeWinter) and connect to projective sets and projective two-weight codes, the work expands the landscape of Denniston-type SRGs. The results set the stage for broader generalizations to non-elementary abelian groups and for discovering new parameter families beyond the classical Latin square types, with potential impact on coding theory and finite geometry.
Abstract
We construct two families of strongly regular Cayley graphs, or equivalently, partial difference sets, based on elementary abelian groups. The parameters of these two families are generalizations of the Denniston and the dual Denniston parameters, in contrast to the well known Latin square type and negative Latin square type parameters. The two families unify and subsume a number of existing constructions which have been presented in various contexts such as strongly regular graphs, partial difference sets, projective sets, and projective two-weight codes, notably including Denniston's seminal construction concerning maximal arcs in classical projective planes with even order. Our construction generates further momentum in this area, which recently saw exciting progress on the construction of the analogue of the famous Denniston partial difference sets in odd characteristic.