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On the non-Abelian group code capacity of memoryless channels

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The author is supported by Fundação Universidade Federal do Pampa - UNIPAMPA, Brazil

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  • 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.

    Mathematics Subject Classification: Primary: 94A05, 94A24; Secondary: 94B60.

    Citation:

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  • Table 1.  $ G(\mathit{\boldsymbol{l}}) $-Symmetric sub-channels the $ D_4 $-symmetric channel 8PSK-AWGN

    $ \rho $ Array Sub-group $ G(\mathit{\boldsymbol{l}}_{ijk}) $ Sub-Constellation $ {\mathcal{X}}(\mathit{\boldsymbol{l}}_{ijk}) $
    1 $\left( {\begin{array}{*{20}{c}} 1&1\\ 0&{} \end{array}} \right)$ $ 2 {\mathbb{Z}}_4\boxtimes \{0\}=\{e,a^2\} $ $ \{x_0,x_4\} $
    $\left( {\begin{array}{*{20}{c}} 1&1\\ 1&{} \end{array}} \right)$ $ 2 {\mathbb{Z}}_4\boxtimes {\mathbb{Z}}_2 = \{e,b,a^2,a^2b\} $ $ \{x_0,x_1,x_4,x_5\} $
    2 $\left( {\begin{array}{*{20}{c}} 1&2\\ 0&{} \end{array}} \right)$ $ {\mathbb{Z}}_4\boxtimes \{0\}=\{e,a,a^2,a^3\} $ $ \{x_0,x_2,x_4,x_6\} $
    $\left( {\begin{array}{*{20}{c}} 1&2\\ 1&{} \end{array}} \right)$ $ {\mathbb{Z}}_4\boxtimes {\mathbb{Z}}_2 = D_4 $ $ {\mathcal{X}}_8 $
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    Table 2.  Output probability densities $ \lambda_{\mathit{\boldsymbol{l}}_{ijk}} $ and capacities $ C_{\mathit{\boldsymbol{l}}_{ijk}} $ of the sub-channels $ G(\mathit{\boldsymbol{l}}) $ of the $ D_4 $-symmetric channel 8PSK-AWGN, where $ p_i(y): = p(y\vert x_i) $

    Array Sub-Constell. $ {\mathcal{X}}(\mathit{\boldsymbol{l}}_{ijk}) $ Density $ \lambda_{\mathit{\boldsymbol{l}}_{ijk}} $ Capacity $ C_{\mathit{\boldsymbol{l}}_{ijk}} $
    $\left( {\begin{array}{*{20}{c}} 1&1\\ 0&{} \end{array}} \right)$ $ \{x_0,x_4\} $ $ \frac{1}{2}(p_0+p_4) $ $ H(\lambda_{\mathit{\boldsymbol{l}}_{110}})-H(p_0) $
    $\left( {\begin{array}{*{20}{c}} 1&1\\ 1&{} \end{array}} \right)$ $ \{x_0,x_1,x_4,x_5\} $ $ \frac{1}{4}(p_0+p_1+p_4+p_5) $ $ H(\lambda_{\mathit{\boldsymbol{l}}_{111}})-H(p_0) $
    $\left( {\begin{array}{*{20}{c}} 1&2\\ 0&{} \end{array}} \right)$ $ \{x_0,x_2,x_4,x_6\} $ $ \frac{1}{4}(p_0+p_2+p_4+p_6) $ $ H(\lambda_{\mathit{\boldsymbol{l}}_{120}})-H(p_0) $
    $\left( {\begin{array}{*{20}{c}} 1&2\\ 1&{} \end{array}} \right)$ $ {\mathcal{X}}_8 $ $ \frac{1}{8}(p_0+p_1+\dots+p_7) $ $ H(\lambda_{\mathit{\boldsymbol{l}}_{121}})-H(p_0) $
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