Group factorizations and information theory

A factorization of a group 9 is a collection of subsets (A(1), A(2),...,A(r)) such that every element g is an element of G has a unique representation g = a(1)center dot a(2)center dot...center dot a(r) where a(i) is an element of A(i) for i = 1,...,r. We shall survey several applications of group factorizations in information theory. They occur in the analysis of syndromes of integer codes, several graphs with large girth important for LDPC codes can be constructed using group factorizations, and various cryptosystems are based on them.