Strongly Regular Graphs with Generalized Denniston and Dual Generalized Denniston Parameters

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.

💡 Research Summary

The paper investigates strongly regular Cayley graphs (SRGs) and their equivalent partial difference sets (PDSs) on elementary abelian groups, introducing two infinite families whose parameters generalize the classical Denniston and dual‑Denniston families. After recalling the basic definitions of SRGs, Cayley graphs, and PDSs, the authors emphasize the well‑known equivalence between a Cayley graph Cay(G,D) being an SRG and D being a (v,k,λ,μ)‑PDS in the additive group G. Lemma 1.9 further connects these objects to projective sets and two‑weight projective codes, establishing a unified framework that links graph theory, finite geometry, and coding theory.

The first main result, Theorem 1.10, constructs, for any prime power q = p^s, any positive integers m and ℓ, and any integer r with 0 ≤ r ≤ m, a PDS D_r in the elementary abelian p‑group
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