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

Denniston [12] constructed partial difference sets (PDS) with parameters (2(3m), (2 (m + r) -2(m) +2 (R))(2(m-1)),2(m)-2 (R)+(2(m+r)- 2(m+2r))(2(r-2)), (2(m+r)-2(m+2r))(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 is an element of {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(3m), ( 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).