Characterising bimodal collections of sets in finite groups

A collection of disjoint subsets A = {A₁, A₂, ..., A_m} of a finite abelian group has the bimodal property if each non-zero group element δ either never occurs as a difference between an element of A_i and an element of A_j with j ≠ i, or else for every element a_i in A_i there is an element a_j ∈ A_j for some j ≠ i with a_i - a_j = δ. 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.