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

Denniston [12] constructed partial difference sets (PDS) with parameters ( 2 3 m , ( 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 3 m for all m ≥ 2 and 1 ≤ 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 3 m , ( 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 3 m for all m ≥ 2 and r ∈ { 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 3 m , ( 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 3 m for all m ≥ 2 and 1 ≤ r < m , where q is an arbitrary prime power.