Skew Hadamard Difference Sets from Dickson Polynomials of Order 7
Skew Hadamard difference sets are an interesting topic of study for over seventy years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in $\mathbb{F}_q$ where $q \equiv 3 \bmod{4}$) were the only example in abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of $\mathcal{D}_5(x^2,u)$ is a new skew Hadamard difference set in $(\mathbb{F}_{3^m},+)$ with $m$ odd, where $\mathcal{D}_n(x,u)$ denotes the first kind of Dickson polynomials of order $n$ and $u \in \mathbb{F}_q^*$. The key observation in the proof is that $\mathcal{D}_5(x^2,u)$ is a planar function from $\mathbb{F}_{3^m}$ to $\mathbb{F}_{3^m}$ 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 $u \in \mathbb{F}_{3^m}^*$, the set $D_u := \{\mathcal{D}_7(x^2,u) : x \in \mathbb{F}_{3^m}^* \}$ is a skew Hadamard difference set in $(\mathbb{F}_{3^m}, +)$, where $m$ is odd and $m \not \equiv 0 \pmod{3}$. The proof is more complicated and different from that of Ding-Yuan skew Hadamard difference sets since $\mathcal{D}_7(x^2,u)$ is not planar in $\mathbb{F}_{3^m}$. 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.