Complete decompositions of finite abelian groups

Abstract

Let G be a nontrivial abelian group and let $A_1,\, \ldots ,\,A_h$ ($h\ge 2$) be nonempty subsets of G. We say that $A_1,\, \ldots ,\,A_h$ is a complete decomposition of G of order h if $A_1+ \cdots +A_h =G$ and $A_i\cap A_j=\emptyset$ for $i,\,j=1,\, \ldots ,\,h$ ($i\ne j$). In this paper we consider the case G is the cyclic group ${\mathbb {Z}}_n$ and determine the values of h for which a complete decomposition of ${\mathbb {Z}}_n$ of order h exists. The result is then extended to the case G is a finite abelian group. We also investigate the existence of complete decompositions of ${\mathbb {Z}}_n$ where the cardinality of each set in the decomposition is a prescribed integer $\ge 2$. As an application, we describe a way to construct codes over a binary alphabet using a construction of a complete decomposition of cyclic groups.