Skew Hadamard difference sets from the Ree–Tits slice symplectic spreads in

Using a class of permutation polynomials of F 3 2 h + 1 obtained from the Ree–Tits slice symplectic spreads in PG ( 3 , 3 2 h + 1 ) , we construct a family of skew Hadamard difference sets in the additive group of F 3 2 h + 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.