Cyclotomic Constructions of Skew Hadamard Difference Sets

TL;DR

This paper presents new constructions of skew Hadamard difference sets in finite fields using cyclotomic classes of specific orders, employing index 2 Gauss sums instead of traditional cyclotomic numbers.

Contribution

It introduces two novel methods for constructing skew Hadamard difference sets using unions of cyclotomic classes of order N=2p_1^m, with a focus on index 2 Gauss sums.

Findings

Constructed skew Hadamard difference sets in finite fields.

Utilized index 2 Gauss sums as main tools.

Provided explicit constructions for specific cyclotomic class orders.

Abstract

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.