Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

TL;DR

This paper constructs new skew Hadamard difference sets in finite fields using Dickson polynomials of order 7, expanding the known family of such sets beyond classical examples.

Contribution

It proves that for all nonzero elements in certain finite fields, the image sets of Dickson polynomials of order 7 form skew Hadamard difference sets, which are inequivalent to known examples.

Findings

New skew Hadamard difference sets constructed using Dickson polynomials of order 7.

Proof that these sets are inequivalent to existing sets for specific field sizes.

Extension of the family of known skew Hadamard difference sets in finite fields.

Abstract

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.