Symplectic spreads and permutation polynomials

2008/10/16 by Simeon Ball, Michael E. Zieve
Mathematics · #math.CO #msc:51E23 #msc:05B25 #msc:11T06

paper · pdf

published as Finite Fields and Applications, Springer Lecture Notes in Computer Science 2948 (2004), 79--88 · 9 pages. This paper was published in 2004. I post it now for greater accessibility

arxiv created 2008/10/16 · arxiv updated 2009/12/01

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(2ⁿ). These new permutation polynomials were later used by Ding, Wang, and Xiang in arXiv:math/0609586 to produce new skew Hadamard difference sets.

Citations

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