Group rings and character sums: tricks of the trade

TL;DR

This paper introduces an accessible overview of techniques combining group rings and character sums to analyze combinatorial structures, providing illustrative examples and clarifying conventions for newcomers.

Contribution

It offers an introductory exposition of the 'tricks of the trade' in using character sums and group rings for combinatorial analysis, filling a gap in existing literature.

Findings

Clarifies conventions and methods used in character sum analysis

Provides illustrative examples for better understanding

Includes explicit proofs of fundamental results

Abstract

The combination of the group ring setting with the methods of character theory allows an elegant and powerful analysis of various combinatorial structures, via their character sums. These combinatorial structures include difference sets, relative difference sets, partial difference sets, bent functions, hyperplanes, spreads, and LP-packings. However, the literature on these techniques often relies on peculiar conventions and implicit understandings that are not always readily accessible to those new to the subject. While there are many excellent advanced sources describing these techniques, we are not aware of an expository paper at the introductory level that articulates the commonly used ``tricks of the trade''. We attempt to remedy this situation by means of illustrative examples, explicit discussion of conventions, and instructive proofs of fundamental results.