Ensembles of codes over abelian groups
Giacomo Como, Fabio Fagnani · 2005
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 Zopfmas a generating group. For codes which are Zopfm-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 2r-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