Characterising bimodal collections of sets in finite groups

Abstract

A collection of disjoint subsets \({\mathcal {A}}=\{A_1,A_2,\ldots ,A_m\}\) of a finite abelian group has the bimodal property if each non-zero group element \(\delta \) either never occurs as a difference between an element of \(A_i\), and an element of \(A_j\) with \(j\ne i\), or else for every element \(a_i\) in \(A_i\), there is an element \(a_j\in A_j\) for some \(j\ne i\) with \(a_i-a_j=\delta \). This property arises in familiar situations, such as cosets of a fixed subgroup or in a group partition, and has applications to the construction of optimal algebraic manipulation detection codes. In this paper, we obtain a structural characterisation for bimodal collections of sets.