Ensembles of codes over abelian groups

In this paper we study ensembles of Abelian group codes on symmetric channels. Our main example is the AWGN channel where the inputs are restricted to a GU constellation admitting Zopf_m as a generating group. For codes which are Zopf_m-free modules, we prove a sort of Shannon theorem which exactly characterizes which are the rates for which reliable transmission is possible. In particular we prove that for the 2^r-PSK constellation, group codes over Zopf- do achieve Shannon capacity. Finally, we study the performance of low density Zopf codes and we prove average convergence rate of their word error probability