Generalized Skew Hadamard Difference Sets

TL;DR

This paper introduces Generalized Skew Hadamard Difference Sets (GSHDS), extending existing theory to cases where p ≡ 1 mod 4, and provides new necessary conditions and divisibility results for their existence.

Contribution

It generalizes skew Hadamard difference sets to include cases with p ≡ 1 mod 4 and extends known existence conditions and bounds to this broader class.

Findings

Extended necessary existence conditions for GSHDS

Derived exponent bounds for GSHDS

Established p-divisibility conditions for difference intersection numbers

Abstract

A skew Hadamard difference set (SHDS) is a difference set that satisfies the skew condition. It is known that if a group G admits a skew hadamard difference set, then G is a p-group with order congruent to 3 modulo 4. We will generalize skew Hadamard difference sets to Generalized Skew Hadamard Difference Sets (GSHDS) to cover the case when p is congruent to 1 module 4, and we will extend all known results that yield necessary existence conditions of skew Hadamard difference sets to our generalization, including the known exponent bounds. We will also show a set of necessary existence conditions a special family of groups. We will close the article with a general p-divisibility condition of the difference intersection numbers of a GSHDS for a special class of subgroups L.