On the non-Abelian group code capacity of memoryless channels

Jorge P. Arpasi

doi:10.3934/amc.2020058

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2020, Volume 14, Issue 3: 423-436. Doi: 10.3934/amc.2020058

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On the non-Abelian group code capacity of memoryless channels

Jorge P. Arpasi^,

Avenida Tiaraju 810, Alegrete, RS, 97541-151, Brazil

^* Corresponding author
^* Corresponding author

Received: October 31, 2018

Revised: July 31, 2019

Early access: January 2020

Published: August 2020

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

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Abstract

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.

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

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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\} $
1	$\left( {\begin{array}{*{20}{c}} 1&1\\ 1&{} \end{array}} \right)$	$ 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\} $
2	$\left( {\begin{array}{*{20}{c}} 1&2\\ 1&{} \end{array}} \right)$	$ {\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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