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February, 1971 Group Codes do not Achieve Shannon's Channel Capacity for General Discrete Channels
R. Ahlswede
Ann. Math. Statist. 42(1): 224-240 (February, 1971). DOI: 10.1214/aoms/1177693508


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.


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R. Ahlswede. "Group Codes do not Achieve Shannon's Channel Capacity for General Discrete Channels." Ann. Math. Statist. 42 (1) 224 - 240, February, 1971.


Published: February, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0219.94008
MathSciNet: MR351615
Digital Object Identifier: 10.1214/aoms/1177693508

Rights: Copyright © 1971 Institute of Mathematical Statistics

Vol.42 • No. 1 • February, 1971
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