Symplectic spreads and permutation polynomials

TL;DR

The paper establishes a link between symplectic spreads in projective geometry and permutation polynomials over finite fields, providing algebraic proofs of known spreads and introducing new low-degree permutation polynomials with applications in combinatorics.

Contribution

It introduces a novel algebraic connection between symplectic spreads and permutation polynomials, including new families of low-degree polynomials and proofs of existing spreads.

Findings

Established a correspondence between symplectic spreads and permutation polynomials.

Provided algebraic proofs for the existence of specific spreads.

Introduced new low-degree permutation polynomials over GF(3^{2h+1}).

Abstract

Every symplectic spread of PG(3,q), or equivalently every ovoid of Q(4,q), is shown to give rise to a certain family of permutation polynomials of GF(q) and conversely. This leads to an algebraic proof of the existence of the Tits-Luneburg spread of W(2^{2h+1}) and the Ree-Tits spread of W(3^{2h+1}), as well as to a new family of low-degree permutation polynomials over GF(3^{2h+1}). We prove the permutation property of the latter polynomials via an odd characteristic analogue of Dobbertin's approach to uniformly representable permutation polynomials over GF(2^n). These new permutation polynomials were later used by Ding, Wang, and Xiang in arXiv:math/0609586 to produce new skew Hadamard difference sets.