Elias ,  proved that group codes achieve Shannon's channel capacity for binary symmetric channels. This result was generalized by Dobrushin  (and independently by Drygas ) 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.
"Group Codes do not Achieve Shannon's Channel Capacity for General Discrete Channels." Ann. Math. Statist. 42 (1) 224 - 240, February, 1971. https://doi.org/10.1214/aoms/1177693508