Partial difference sets with Denniston parameters in elementary abelian p-groups

Abstract

Denniston [12] constructed partial difference sets (PDS) with parameters $(2^{3m},(2^{m+r}-2^m+2^r\left)\right(2^m-1),2^m-2^r+(2^{m+r}-2^m+2^r\left)\right(2^r-2),(2^{m+r}-2^m+2^r\left)\right(2^r-1\left)\right)$ in elementary abelian groups of order $2^{3m}$ for all $m\ge 2$ and $1\le r<m$. These PDS arise from maximal arcs in the Desarguesian projective planes PG$(2,2^m)$. Davis et al. [10] and also De Winter [13] presented constructions of PDS with Denniston parameters $(p^{3m},(p^{m+r}-p^m+p^r\left)\right(p^m-1),p^m-p^r+(p^{m+r}-p^m+p^r\left)\right(p^r-2),(p^{m+r}-p^m+p^r\left)\right(p^r-1\left)\right)$ in elementary abelian groups of order $p^{3m}$ for all $m\ge 2$ and $r\in \{1,m-1\}$, where p is an odd prime. The constructions in [10], [13] are particularly intriguing, as it was shown by Ball, Blokhuis, and Mazzocca [1] 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 $(q^{3m},(q^{m+r}-q^m+q^r\left)\right(q^m-1),q^m-q^r+(q^{m+r}-q^m+q^r\left)\right(q^r-2),(q^{m+r}-q^m+q^r\left)\right(q^r-1\left)\right)$ exist in elementary abelian groups of order $q^{3m}$ for all $m\ge 2$ and $1\le r<m$, where q is an arbitrary prime power.