Group Codes do not Achieve Shannon's Channel Capacity for General Discrete Channels

Abstract

Elias [9], [10] proved that group codes achieve Shannon's channel capacity for binary symmetric channels. This result was generalized by Dobrushin [7] (and independently by Drygas [8]) to discrete memoryless channels satisfying a certain symmetry condition and having a Galois field as alphabet. We prove that group codes to dnot achieve the channel capacity for general discrete memoryless channels. It therefore makes sense to introduce a group code capacity and to talk about a group coding theorem and its weak converse can be established for several reasonable channels such as the discrete memoryless channel, compound channels, and averaged channels. An example of a channel is given for which Shannon's capacity is positive and the group code capacity is zero. Using group codes, one can therefore expect high rates only for channels with a simple probabilistic structure.