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

Denniston {D1969} constructed partial difference sets (PDS) with parameters (2³ᵐ, (2ᵐ⁺ʳ-2ᵐ+2ʳ)(2ᵐ-1), 2ᵐ-2ʳ+(2ᵐ⁺ʳ-2ᵐ+2ʳ)(2ʳ-2), (2ᵐ⁺ʳ-2ᵐ+2ʳ)(2ʳ-1)) in elementary abelian groups of order 2³ᵐ 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 (p³ᵐ, (pᵐ⁺ʳ-pᵐ+pʳ)(pᵐ-1), pᵐ-pʳ+(pᵐ⁺ʳ-pᵐ+pʳ)(pʳ-2), (pᵐ⁺ʳ-pᵐ+pʳ)(pʳ-1)) in elementary abelian groups of order p³ᵐ 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 (q³ᵐ, (qᵐ⁺ʳ-qᵐ+qʳ)(qᵐ-1), qᵐ-qʳ+(qᵐ⁺ʳ-qᵐ+qʳ)(qʳ-2), (qᵐ⁺ʳ-qᵐ+qʳ)(qʳ-1)) exist in elementary abelian groups of order q³ᵐ for all m ≥ 2 and 1 ≤ r < m, where q is an arbitrary prime power.