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: November 2018

Revised: August 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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References

[1]	R. Ahlswede, Group codes do not achieve shannon's channel capacity for general discrete channels, The Annals of Mathematical Statistics, 42 (1971), 224-240. doi: 10.1214/aoms/1177693508.
[2]	J. P. Arpasi, One example of a non-Abelian group code over AWGN channels, Proceedings of the 14th Canadian Workshop on Information Theory, (2015), 115–119. doi: 10.1109/CWIT.2015.7255165.
[3]	G. Como, Group codes outperform binary-coset codes on non-binary memoryless channels, IEEE Trans. Inform. Theory, 56 (2010), 4321-4334. doi: 10.1109/TIT.2010.2054330.
[4]	G. Como and F. Fagnani, The capacity of abelian group codes over symmetric channels, IEEE Trans. Inform. Theory, 45 (2009), 3-31.
[5]	G. Como and F. Fagnani, Average spectra and minimum distance of low-density parity-check codes over abelian groups, SIAM J. Discrete Math., 23 (2008/09), 19-53. doi: 10.1137/070686615.
[6]	T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edition, Wiley InterScience, Piscataway, NJ, 2006.
[7]	G. D. Forney, Geometrically uniform codes, IEEE Trans. Inform. Theory, 37 (1991), 1241-1260. doi: 10.1109/18.133243.
[8]	R. Gallager, Information Theory and Reliable Communication, Wiley and Sons, 1970. doi: 10.1007/978-3-7091-2945-6.
[9]	F. Garin and F. Fagnani, Analysis of serial turbo codes over abelian groups for symmetric channels, SIAM J. Discrete Math., 22 (2008), 1488-1526. doi: 10.1137/07068802X.
[10]	I. N. Herstein, Topics in Algebra, 2nd edition, Wiley and Sons, New York, 1975.
[11]	H. J. Kim, J. B. Nation and A. V. Shepler, Group coding with complex isometries, IEEE Trans. Inform. Theory, 61 (2015), 33-50. doi: 10.1109/TIT.2014.2365020.
[12]	H. A. Loeliger, Signal sets matched to groups, IEEE Trans. Inform. Theory, 37 (1991), 1675-1682. doi: 10.1109/18.104333.
[13]	H. A. Loeliger and T. Mittelholzer, Convolutional codes over groups. Codes and complexity, IEEE Trans. Inform. Theory, 42 (1996), 1660-1686. doi: 10.1109/18.556664.
[14]	T. Mittelholzer and J. Lahtonen, Group codes generated by finite reflection groups, IEEE Trans. Inform. Theory, 42 (1996), 519-528. doi: 10.1109/18.485721.
[15]	W. W. Peterson, J. B. Nation and M. P. Fossorier, Reflection group codes and their decoding, IEEE Trans. Inform. Theory, 56 (2010), 6273-6293. doi: 10.1109/TIT.2010.2080571.
[16]	J. J. Rotman, An Introduction to the Theory of the Groups, 4th edition, Graduate Texts in Mathematics, 148. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4176-8.
[17]	A. G. Sahebi and P. S. Pradhan, Abelian group codes for channel coding and source coding, IEEE Trans. Inform. Theory, 61 (2015), 2399-2414. doi: 10.1109/TIT.2015.2407874.
[18]	N. Shulman and M. Feder, Random coding techniques for non-random codes, IEEE Trans. Inform. Theory, 45 (1999), 2101-2104. doi: 10.1109/18.782147.
[19]	D. Slepian, Group codes for the gaussian channels, Bell Systems Technical Journal, 47 (1968), 575-602. doi: 10.1002/j.1538-7305.1968.tb02486.x.