Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3,3^{2h+1})

TL;DR

This paper constructs new skew Hadamard difference sets in finite fields using permutation polynomials derived from Ree-Tits symplectic spreads, and explores their properties and relations to twin prime power difference sets.

Contribution

It introduces a novel family of skew Hadamard difference sets from Ree-Tits symplectic spreads and analyzes their inequivalence and connection to twin prime power difference sets.

Findings

New skew Hadamard difference sets for h=2,3

Conjecture of novelty for all h>3

Inequivalence with classical difference sets

Abstract

Using a class of permutation polynomials of $F_{3^{2h+1}}$ obtained from the Ree-Tits symplectic spreads in $PG(3,3^{2h+1})$, we construct a family of skew Hadamard difference sets in the additive group of $F_{3^{2h+1}}$. With the help of a computer, we show that these skew Hadamard difference sets are new when $h=2$ and $h=3$. We conjecture that they are always new when $h>3$. Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime power parameters.