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.