Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes

Yani Zhang; Xiaomin Bao; Zhihua Yuan; Xusheng Wu

doi:doi:10.11648/j.ajmcm.20170201.12

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Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes

Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu

Published in American Journal of Mathematical and Computer Modelling (Volume 2, Issue 1)

Received: 29 October 2016 Accepted: 28 November 2016 Published: 6 January 2017

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Abstract

In this paper, a new efficient syndrome-weight decoding algorithm (NESWDA) is presented to decode up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of NESWDA is based on the property cyclic codes together with the weight of syndrome difference. The advantage of the NESWDA decoding algorithm over the previous table look-up methods is that it has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes.

Published in	American Journal of Mathematical and Computer Modelling (Volume 2, Issue 1)
DOI	10.11648/j.ajmcm.20170201.12
Page(s)	6-12
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Cyclic Codes, Decoding, Quadratic Residue Code

References

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[3]	Y. H. Chen, T. K. Truong, C. H. Huang, C. H. Chien, A lookup table decoding of systematic (47, 24, 11)quadratic residue code, Information Science 179 (2009) 2470-2477.
[4]	T. C. Lin, H. P. Lee, H. C. Chang, S. I. Chu, T. K. Truong, High speed decoding of the binary (47, 24, 11)quadratic residue code, Information Science 180 (2010) 4060-4068.
[5]	T. C. Lin, H. P. Lee, H. C. Chang, T. K. Truong, A cyclic weight algorithm of decoding (47, 24, 11) quadratic residue code, Information Science 197 (2012) 215-222.
[6]	G. Dubney, I. S. Reed, T. K. Truong, J. Yang, Decoding the (47, 24, 11) quadratic residue code using bit-error probability estimates, IEEE Transactions on Communications 57 (2009).
[7]	R. He, I. S. Reed, T. K. Truong, X. Chen, Decoding the (47, 24, 11) quadratic residue code, IEEE Transactions on Information Theory 47 (2001) 1181-1186.
[8]	T. C. Lin, T. K. Truong, and H. P. Lee, Algebraic decoding of the (41, 21, 9) quadratic residue code, Information Sciences 179 (2009) 3451-3459.
[9]	X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong, Use of Grobner bases to decode binary cyclic codes up to the true minimum distance, IEEE Transactions on Information Theory 40 (1994) 1654-1661.
[10]	Y. H. Chen, T. K. Truong, Y. Chang, C. D. Lee, S. H. Chen, Algebraic decoding of Quadratic Residue codes using Berlekamp–Massey algorithm, Journal of Information Science and Engineering 23 (2007) 127-145.
[11]	I. S. Reed, X. Chen, Error-Control Coding for Data Networks, Kluwer, Boston, MA, 1999.
[12]	I. S. Reed, M. T. Shih, T. K. Truong, VLSI design of inverse-free Berlekamp-Massey algorithm, Processings IEE 138 (1991) 295-298.
[13]	I. S. Reed, T. K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38(1992) 974-986.
[14]	I. S. Reed, X. Yin, T. K. Truong, Algebraic decoding of the (32, 16, 8) Quadratic Residue code, IEEE Transactions on Information Theory 36 (1990) 876-880.
[15]	I. S. Reed, X. Yin, T. K. Truong, J. K. Holmes, Decoding the (24, 12, 8) Golay code, IEE Proceedings 137 (3) (1990) 202-206.

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APA Style

Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu. (2017). Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. American Journal of Mathematical and Computer Modelling, 2(1), 6-12. https://doi.org/10.11648/j.ajmcm.20170201.12

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ACS Style

Yani Zhang; Xiaomin Bao; Zhihua Yuan; Xusheng Wu. Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. Am. J. Math. Comput. Model. 2017, 2(1), 6-12. doi: 10.11648/j.ajmcm.20170201.12

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AMA Style

Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu. Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. Am J Math Comput Model. 2017;2(1):6-12. doi: 10.11648/j.ajmcm.20170201.12

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@article{10.11648/j.ajmcm.20170201.12,
  author = {Yani Zhang and Xiaomin Bao and Zhihua Yuan and Xusheng Wu},
  title = {Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {1},
  pages = {6-12},
  doi = {10.11648/j.ajmcm.20170201.12},
  url = {https://doi.org/10.11648/j.ajmcm.20170201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170201.12},
  abstract = {In this paper, a new efficient syndrome-weight decoding algorithm (NESWDA) is presented to decode up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of NESWDA is based on the property cyclic codes together with the weight of syndrome difference. The advantage of the NESWDA decoding algorithm over the previous table look-up methods is that it has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes.},
 year = {2017}
}

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TY  - JOUR
T1  - Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes
AU  - Yani Zhang
AU  - Xiaomin Bao
AU  - Zhihua Yuan
AU  - Xusheng Wu
Y1  - 2017/01/06
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170201.12
DO  - 10.11648/j.ajmcm.20170201.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 6
EP  - 12
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170201.12
AB  - In this paper, a new efficient syndrome-weight decoding algorithm (NESWDA) is presented to decode up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of NESWDA is based on the property cyclic codes together with the weight of syndrome difference. The advantage of the NESWDA decoding algorithm over the previous table look-up methods is that it has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes.
VL  - 2
IS  - 1
ER  -

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Author Information

Yani Zhang

School of Mathematics & Statistics, Southwest University, Chongqing, China
Xiaomin Bao

School of Mathematics & Statistics, Southwest University, Chongqing, China
Zhihua Yuan

School of Mathematics & Statistics, Southwest University, Chongqing, China
Xusheng Wu

School of Mathematics & Statistics, Southwest University, Chongqing, China

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APA Style

Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu. (2017). Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. American Journal of Mathematical and Computer Modelling, 2(1), 6-12. https://doi.org/10.11648/j.ajmcm.20170201.12

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ACS Style

Yani Zhang; Xiaomin Bao; Zhihua Yuan; Xusheng Wu. Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. Am. J. Math. Comput. Model. 2017, 2(1), 6-12. doi: 10.11648/j.ajmcm.20170201.12

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AMA Style

Yani Zhang, Xiaomin Bao, Zhihua Yuan, Xusheng Wu. Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes. Am J Math Comput Model. 2017;2(1):6-12. doi: 10.11648/j.ajmcm.20170201.12

Copy | Download

@article{10.11648/j.ajmcm.20170201.12,
  author = {Yani Zhang and Xiaomin Bao and Zhihua Yuan and Xusheng Wu},
  title = {Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {1},
  pages = {6-12},
  doi = {10.11648/j.ajmcm.20170201.12},
  url = {https://doi.org/10.11648/j.ajmcm.20170201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170201.12},
  abstract = {In this paper, a new efficient syndrome-weight decoding algorithm (NESWDA) is presented to decode up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of NESWDA is based on the property cyclic codes together with the weight of syndrome difference. The advantage of the NESWDA decoding algorithm over the previous table look-up methods is that it has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Decoding of the Five-Error-Correcting Binary Quadratic Residue Codes
AU  - Yani Zhang
AU  - Xiaomin Bao
AU  - Zhihua Yuan
AU  - Xusheng Wu
Y1  - 2017/01/06
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170201.12
DO  - 10.11648/j.ajmcm.20170201.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 6
EP  - 12
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170201.12
AB  - In this paper, a new efficient syndrome-weight decoding algorithm (NESWDA) is presented to decode up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of NESWDA is based on the property cyclic codes together with the weight of syndrome difference. The advantage of the NESWDA decoding algorithm over the previous table look-up methods is that it has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. Moreover, it can be extended to decode all five-error-correcting binary QR codes.
VL  - 2
IS  - 1
ER  -

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