Difference Sets with Few Character Values

TL;DR

This paper investigates difference sets with few character values, focusing on those with gcd$(v,n)>1$, and explores conditions for difference sets with three nontrivial character values to advance understanding of their structure.

Contribution

It introduces necessary conditions for difference sets with three nontrivial character values, aiming to find examples that do not satisfy the character divisibility property.

Findings

Identified conditions for difference sets with three nontrivial character values.

Provided insights into the structure of difference sets with gcd$(v,n)>1$.

Addressed the open problem posed by Jungnickel and Schmidt.

Abstract

The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd$(v,n)>1$. It is remarkable that all the known difference sets with gcd$(v,n)>1$ have the so-called character divisibility property. In 1997, Jungnickel and Schmidt posed the problem of constructing difference sets with gcd$(v,n)>1$ that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.