On the existence and construction of modular difference sets

Abstract

The concept of a modular difference set was originally motivated by the cognate notion of modular Hadamard matrices, which have been researched extensively. We initiate the study of the repetition-parameter set in a modular difference set, and we relate the repetition-parameter set to integer partitions and Diophantine equations. By example, we show how a computational study of integer partitions can improve the upper bound on the size of such repetition-parameter set. All previously known examples of modular difference sets in a direct product of groups are concerned with a product of just two groups. We present new constructions of modular difference sets in a direct product of n groups. These new constructions suggest that the size of the repetition-parameter set is intimately related to the group’s structure. A generalization of difference sets, partial difference sets, and relative difference sets, modular difference sets have been used to construct both modular symmetric designs and equiangular tight frames in finite fields.