The capacity of Abelian group codes over symmetric channels

In this paper the capacity achievable by Abelian group codes when employed over symmetric channels is determined. For certain important examples, like the AWGN channel with m-PSK modulation, it follows that this capacity coincides with the corresponding Shannon capacity of these channels. In other words using Abelian group codes in this case there is no loss of capacity (as it happens for binary linear codes for binary symmetric channels). Finally, a three dimensional modulation is presented for which instead, despite its group symmetry, the use of Abelian group codes leads to a loss in capacity.