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 F-q where q equivalent to 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 D-5(x(2),u) is a new skew Hadamard difference set in (F-3m,+) with m odd, where D-n(x,u) denotes the first kind of Dickson polynomials of order n and u is an element of F-q*. The key observation in the proof is that D-5(x(2),u) is a planar function from F-3m to F-3m 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 is an element of F-3m* , the set D-u:= {D-7(x(2), u): x is an element of F-3m*} is a skew Hadamard difference set in (F-3m,+), where m is odd and m not equal 0 (mod 3). The proof is more complicated and different than that of Ding-Yuan skew Hadamard difference sets since D-7(x(2),u) is not planar in F-3m. 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. (C) 2014 Wiley Periodicals, Inc.