Skew Hadamard difference sets from the Ree–Tits slice symplectic spreads in PG(3,32h+1)

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Abstract

Using a class of permutation polynomials of F32h+1 obtained from the Ree–Tits slice symplectic spreads in PG(3,32h+1), we construct a family of skew Hadamard difference sets in the additive group of F32h+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.

Keywords

Difference set
Gauss sum
Permutation polynomial
Ree–Tits slice spread
Skew Hadamard difference set
Symplectic spread
Twin prime power difference set

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Research supported in part by NSF Grant DMS 0400411.