Partial Difference Sets with Denniston Parameters in Elementary Abelian $p$-Groups

Abstract

Denniston \cite{D1969} constructed partial difference sets (PDS) with parameters $(2^{3m}, (2^{m+r}-2^m+2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m\geq 2$ and $1 \leq r < m$. These PDS correspond to maximal arcs in the Desarguesian projective planes PG$(2, 2^m)$. Davis et al. \cite{DHJP2024} and also De Winter \cite{dewinter23} presented constructions of PDS with Denniston parameters $(p^{3m}, (p^{m+r}-p^m+p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ in elementary abelian groups of order $p^{3m}$ for all $m \geq 2$ and $r \in \{1, m-1\}$, where $p$ is an odd prime. The constructions in \cite{DHJP2024, dewinter23} are particularly intriguing, as it was shown by Ball, Blokhuis, and Mazzocca \cite{BBM1997} that no nontrivial maximal arcs in PG$(2, q^m)$ exist for any odd prime power $q$. In this paper, we show that PDS with Denniston parameters $(q^{3m}, (q^{m+r}-q^m+q^r)(q^m-1), q^m-q^r+(q^{m+r}-q^m+q^r)(q^r-2), (q^{m+r}-q^m+q^r)(q^r-1))$ exist in elementary abelian groups of order $q^{3m}$ for all $m \geq 2$ and $1 \leq r < m$, where $q$ is an arbitrary prime power.