Characterising bimodal collections of sets in finite groups

A collection of disjoint subsets A = {A1, A2, ..., Am} 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 Ai and an element of Aj with j ≠ i, or else for every element ai in Ai there is an element aj ∈ Aj for some j ≠ i with ai - aj = 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