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On the non-Abelian group code capacity of memoryless channels
Avenida Tiaraju 810, Alegrete, RS, 97541-151, 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.
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. |
show all references
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. |
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