Characterising bimodal collections of sets in finite groups

collection of disjoint subsets 𝒜={A_1,A_2,… ,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.