A new structure for difference matrices over abelian pp-groups

A difference matrix over a group is a discrete structure that is intimately related to many other combinatorial designs, including mutually orthogonal Latin squares, orthogonal arrays, and transversal designs. Interest in constructing difference matrices over $2$-groups has been renewed by the recent discovery that these matrices can be used to construct large linking systems of difference sets, which in turn provide examples of systems of linked symmetric designs and association schemes. We survey the main constructive and nonexistence results for difference matrices, beginning with a classical construction based on the properties of a finite field. We then introduce the concept of a contracted difference matrix, which generates a much larger difference matrix. We show that several of the main constructive results for difference matrices over abelian $p$-groups can be substantially simplified and extended using contracted difference matrices. In particular, we obtain new linking systems of difference sets of size $7$ in infinite families of abelian $2$-groups, whereas previously the largest known size was $3$.Comment: 27 pages. Discussion of new reference [LT04

Group rings and character sums: tricks of the trade

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, 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.Comment: 22 page