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

TL;DR

This paper generalizes the construction of partial difference sets with Denniston parameters to all prime powers, expanding their existence beyond previously known cases and linking them to maximal arcs in finite geometries.

Contribution

It proves the existence of PDS with Denniston parameters in elementary abelian groups for all prime powers, broadening the scope of known constructions.

Findings

PDS with Denniston parameters exist for all prime powers q.

Construction applies to all m ≥ 2 and 1 ≤ r < m.

Links to maximal arcs in finite projective planes.

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.