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 ≡ 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 ofD 5 ( x 2 , u )is a new skew Hadamard difference set in ( F 3 m , + ) with m odd, whereD n ( x , u )denotes the first kind of Dickson polynomials of order n and u ∈ F q * . The key observation in the proof is thatD 5 ( x 2 , u )is a planar function from F 3 mto F 3 mfor 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 ∈ F 3 m * , the setD u : = { D 7 ( x 2 , u ) : x ∈ F 3 m * }is a skew Hadamard difference set in ( F 3 m , + ) , where m is odd andm ¬ ≡ 0( mod 3 ) . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets sinceD 7 ( x 2 , u )is not planar in F 3 m . 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.