The capacity of finite Abelian group codes over symmetric memoryless channels

Abstract

The capacity of finite Abelian group codes over symmetric memoryless channels is determined. For certain important examples, such as m -PSK constellations over additive white Gaussian noise (AWGN) channels, with m a prime power, it is shown that this capacity coincides with the Shannon capacity; i.e., there is no loss in capacity using group codes. (This had previously been known for binary-linear codes used over binary-input output-symmetric memoryless channels.) On the other hand, a counterexample involving a three-dimensional geometrically uniform constellation is presented in which the use of Abelian group codes leads to a loss in capacity. The error exponent of the average group code is determined, and it is shown to be bounded away from the random-coding error exponent, at low rates, for finite Abelian groups not admitting Galois field structure.