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.

中文翻译：

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关于无记忆通道的非阿贝尔群码容量

在这项工作中，提供了通过对称信道传输的非阿贝尔组码的组编码能力$ C_G $的定义。示出了该$ C_G $是允许可靠传输的这些非阿贝尔群码的速率集合的上限。另外，可以推断出$ C_G $是信道容量的下限。此后，计算通过8PSK-AWGN信道发送的二面体组上的组代码的$ C_G $，然后表明它等于信道容量。是否存在非阿贝尔群码的比率任意接近$ C_G $和任意小的错误概率仍然是一个悬而未决的问题。