# American Institute of Mathematical Sciences

August  2020, 14(3): 423-436. doi: 10.3934/amc.2020058

## On the non-Abelian group code capacity of memoryless channels

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* Corresponding author

Received  November 2018 Revised  August 2019 Published  January 2020

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

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.

Citation: Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058
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##### References:
$G(\mathit{\boldsymbol{l}})$-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 $\left( {\begin{array}{*{20}{c}} 1&2\\ 0&{} \end{array}} \right)$ ${\mathbb{Z}}_4\boxtimes \{0\}=\{e,a,a^2,a^3\}$ $\{x_0,x_2,x_4,x_6\}$ $\left( {\begin{array}{*{20}{c}} 1&2\\ 1&{} \end{array}} \right)$ ${\mathbb{Z}}_4\boxtimes {\mathbb{Z}}_2 = D_4$ ${\mathcal{X}}_8$
 $\rho$ 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 $\left( {\begin{array}{*{20}{c}} 1&2\\ 0&{} \end{array}} \right)$ ${\mathbb{Z}}_4\boxtimes \{0\}=\{e,a,a^2,a^3\}$ $\{x_0,x_2,x_4,x_6\}$ $\left( {\begin{array}{*{20}{c}} 1&2\\ 1&{} \end{array}} \right)$ ${\mathbb{Z}}_4\boxtimes {\mathbb{Z}}_2 = D_4$ ${\mathcal{X}}_8$
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 $C_{\mathit{\boldsymbol{l}}_{ijk}}$ $\left( {\begin{array}{*{20}{c}} 1&1\\ 0&{} \end{array}} \right)$ $\{x_0,x_4\}$ $\frac{1}{2}(p_0+p_4)$ $H(\lambda_{\mathit{\boldsymbol{l}}_{110}})-H(p_0)$ $\left( {\begin{array}{*{20}{c}} 1&1\\ 1&{} \end{array}} \right)$ $\{x_0,x_1,x_4,x_5\}$ $\frac{1}{4}(p_0+p_1+p_4+p_5)$ $H(\lambda_{\mathit{\boldsymbol{l}}_{111}})-H(p_0)$ $\left( {\begin{array}{*{20}{c}} 1&2\\ 0&{} \end{array}} \right)$ $\{x_0,x_2,x_4,x_6\}$ $\frac{1}{4}(p_0+p_2+p_4+p_6)$ $H(\lambda_{\mathit{\boldsymbol{l}}_{120}})-H(p_0)$ $\left( {\begin{array}{*{20}{c}} 1&2\\ 1&{} \end{array}} \right)$ ${\mathcal{X}}_8$ $\frac{1}{8}(p_0+p_1+\dots+p_7)$ $H(\lambda_{\mathit{\boldsymbol{l}}_{121}})-H(p_0)$
 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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