Characterising bimodal collections of sets in finite groups

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