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

Denniston D1969 constructed partial difference sets (PDS) with parameters (2^3m, (2^m+r-2ᵐ+2ʳ) (2ᵐ-1), 2ᵐ-2ʳ+ (2^m+r-2ᵐ+2ʳ) (2ʳ-2), (2^m+r-2ᵐ+2ʳ) (2ʳ-1) ) in elementary abelian groups of order 2^3m for all m 2 and 1 r < m. These PDS correspond to maximal arcs in the Desarguesian projective planes PG (2, 2ᵐ). Davis et al. DHJP2024 and also De Winter dewinter23 presented constructions of PDS with Denniston parameters (p^3m, (p^m+r-pᵐ+pʳ) (pᵐ-1), pᵐ-pʳ+ (p^m+r-pᵐ+pʳ) (pʳ-2), (p^m+r-pᵐ+pʳ) (pʳ-1) ) in elementary abelian groups of order p^3m for all m 2 and r \1, m-1\, where p is an odd prime. The constructions in DHJP2024, dewinter23 are particularly intriguing, as it was shown by Ball, Blokhuis, and Mazzocca BBM1997 that no nontrivial maximal arcs in PG (2, qᵐ) exist for any odd prime power q. In this paper, we show that PDS with Denniston parameters (q^3m, (q^m+r-qᵐ+qʳ) (qᵐ-1), qᵐ-qʳ+ (q^m+r-qᵐ+qʳ) (qʳ-2), (q^m+r-qᵐ+qʳ) (qʳ-1) ) exist in elementary abelian groups of order q^3m for all m 2 and 1 r < m, where q is an arbitrary prime power.