Denniston partial difference sets exist in the odd prime case

Abstract

Denniston constructed partial difference sets (PDSs) with the 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 $m\ge 2,1\le r<m$. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters $(p^{3m},(p^{m+r}-p^m+p^r)(p^m-1),p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2),(p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m\ge 2,r\in \{1,m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.

Since our work was announced, we have become aware that an equivalent result has simultaneously been proved by de Winter for projective two-weight sets, and a corresponding coding theory result was proved by Bierbrauer and Edel in 1997; references and citations have been added for these