On the non-Abelian group code capacity of memoryless channels

Abstract

In this work is provided a definition of group encoding capacity $ C_G $ of non-Abelian group codes transmitted through symmetric channels. It is shown that this $ C_G $ is an upper bound of the set of rates of these non-Abelian group codes that allow reliable transmission. Also, is inferred that the $ C_G $ is a lower bound of the channel capacity. After that, is computed the $ C_G $ of the group code over the dihedral group transmitted through the 8PSK-AWGN channel then is shown that it equals the channel capacity. It remains an open problem whether there exist non-Abelian group codes of rate arbitrarily close to $ C_G $ and arbitrarily small error probability.