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

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.