Group rings and character sums: tricks of the trade

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.