Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in Fq where q ≡ 3 mod 4) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of D5(x2, u) is a new skew Hadamard difference set in (F3m ,+) with m odd, where Dn(x, u) denotes the first kind of Dickson polynomials of order n and u ∈ Fq . The key observation in the proof is that D5(x2, u) is a planar function from F3m to F3m for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all u ∈ F3m , the set Du := {D7(x2, u) : x ∈ F3m } is a skew Hadamard difference set in (F3m ,+), where m is odd and m ≡ 0 (mod 3). The proof is more complicated and different than that of Ding-Yuan skew Hadamard difference sets since D7(x2, u) is not planar in F3m . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for m = 5, 7 by comparing the triple intersection numbers. © 2014 Wiley Periodicals, Inc. J. Combin. Designs 23: 436–461, 2015