A new structure for difference matrices over abelian $p$-groups

TL;DR

This paper introduces contracted difference matrices over abelian p-groups, simplifying their construction and enabling the creation of larger linking systems of difference sets, notably increasing the known size to 7.

Contribution

It presents the concept of contracted difference matrices, extending existing constructions and significantly improving the size of known linking systems over abelian 2-groups.

Findings

New construction of linking systems of difference sets of size 7

Simplification of difference matrix constructions over abelian p-groups

Extension of known results using contracted difference matrices

Abstract

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 $3$.