Denniston partial difference sets exist in the odd prime case

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 ≥ 2, 1 ≤ 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 ≥ 2, r ∈ {1, m − 1} where p is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.