Cyclotomy, difference sets, sequences with low correlation, strongly regular graphs, and related geometric substructures

TL;DR

This paper surveys the construction and nonexistence of combinatorial and geometric structures derived from cyclotomic classes, including difference sets, sequences with low correlation, strongly regular graphs, and related substructures in finite fields.

Contribution

It provides a comprehensive overview of classical and recent results on cyclotomic-based difference sets, sequences, and strongly regular graphs, highlighting new constructions and nonexistence proofs.

Findings

Survey of classical and recent results on cyclotomy-based difference sets

Overview of constructions of sequences with low correlation from cyclotomic classes

Compilation of recent results on strongly regular Cayley graphs and geometric substructures

Abstract

In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on difference sets related to cyclotomy, and cyclotomic constructions of sequences with low correlation. We also give an extensive survey of recent results on constructions of strongly regular Cayley graphs and related geometric substructures such as $m$-ovoids and $i$-tight sets in classical polar spaces.