Cyclotomic difference sets in finite fields

TL;DR

This paper establishes new conditions for cyclotomic difference sets in finite fields, solves the problem for even q, and extends nonexistence results for even m up to 22, advancing classification efforts.

Contribution

It introduces new necessary and sufficient conditions using polynomial equations on Gauss sums, including solutions for previously neglected cases.

Findings

Solved the difference set problem for even q

Extended nonexistence results for even m up to 22

Proposed conjectures for full classification

Abstract

The classical problem of whether $m$th-powers with or without zero in a finite field $\mathbb{F}_q$ form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this paper new necessary and sufficient conditions are established including those via a system of polynomial equations on Gauss sums. The author thereby solves the problem for even $q$ which is neglected in the literature, and extends the nonexistence list for even $m$ up to $22$. Moreover, conjectures toward the complete classification are posed.