Characterising bimodal collections of sets in finite groups

Huczynska, S. and Paterson, Maura B. (2019) Characterising bimodal collections of sets in finite groups. Archiv der Mathematik 113 , pp. 571-580. ISSN 0003-889X.

Abstract

A collection of disjoint subsets ${\mathcal A}=\{A_1,A_2,\dotsc,A_m\}$ 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 $j\neq i$, or else for every element $a_i$ in $A_i$ there is an element $a_j\in A_j$ for some $j\neq i$ 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.

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