A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes

Takayasu Kaida; Junru Zheng

doi:doi:10.11648/j.pamj.s.2015040201.17

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A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes

Takayasu Kaida, Junru Zheng

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 2-1)

Received: 22 December 2014 Accepted: 30 January 2015 Published: 11 February 2015

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Abstract

The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 2-1) This article belongs to the Special Issue Abridging over Troubled Water---Scientific Foundation of Engineering Subjects
DOI	10.11648/j.pamj.s.2015040201.17
Page(s)	36-41
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), 2015. Published by Science Publishing Group

Keywords

Cyclic Code, Discrete Fourier Transform, Independent Set, Proposed Algorithm, Recurrent Algorithm, Shift Bound

References

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[2]	C.R.P.Hartmann, K.K.Tzeng, “Generalization of the BCH bound", Information and Control, Vol.20, pp.489-498, 1972.
[3]	T.Kaida, J.Zheng, “A decoding method up to the Hartmann-Tzeng bound using DFT for cyclic codes", Proceedings of 2007 Asia-Pacific Conference on Communications, pp.114-117, 2007.
[4]	T.Kaida, J.Zheng, “A constructing approach of the Roos bound for cyclic codes", Proceedings of 2008 International Symposium on Information Theory and its Applications, pp.395-399, 2008.
[5]	T.Kaida, J.Zheng, “On improved algorithms of the proposed lower bound including well-known bounds for cyclic codes", Proceedings of 2012 International Symposium on Information Theory and its Applications, pp.446-449, 2012.
[6]	T.Kaida, J.Zheng, “On some algorithms on the proposed lower bound for for cyclic codes", Proceedings of 2013 IEEE Asia-Pacific Conference on Comunications, pp.526-530, 2013.
[7]	F.J.MacWilliams, N.J.A.Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.
[8]	J. L. Massey, “Shift-register synthesis and BCH decoding", IEEE Transaction on Information Theory, vol.IT-15, pp.122-127, Jan., 1969.
[9]	R.Pellikaan, “The shift bound for cyclic, Reed-Muller and geometric Goppa codes", Arithmetic, Geometry and Coding Theory 4, pp.155-174, Walter de Gruyter & Co, Berlin, 1996.
[10]	W.W.Peterson, E.J. Weldon, Jr., “Error Correcting Codes", nd ed. Cambridge, MA: MIT Press, 1972.
[11]	F.Ponchio, M.Sala, “A lower bound on the distance of cyclic codes", BCRI Preprint 2003, the file of BCRI-07.ps on http://www.bcri.ucc.ie/nonPeerReviewed.html.
[12]	G.Promhouse and S.E.Tavares, “The minimum distance of all binary cyclic codes of odd lengths from 69 to 99", IEEE Transaction on Information Theory, vol.IT-24, No.4, pp.438-442, July, 1978.
[13]	C.Roos, “A new lower bound for the minimum distance of a cyclic code", IEEE Transaction on Information Theory, Vol.29, No.3, pp.330-332, 1983.
[14]	M.van Eupen, J.H.van Lint, “On the minimum distance of ternary cyclic codes", IEEE Transaction on Information Theory, Vol.39, No.2, pp.409-422, 1993.
[15]	J.H.van Lint, R.M.Wilson, “On the minimum distance of cyclic codes", IEEE Transaction on Information Theory, Vol.32, No.1, pp.23-40, 1986.
[16]	J. Zheng, T. Kaida, “On linear complexity and minimum distance for cyclic codes by defining sequence with unknown elements", The Second International Workshop on Sequence Design and its Application in Communication, CD-ROM No.55, 2005.
[17]	J.Zheng, T.Kaida, “On shift bound for cyclic codes by DFT with unknown elements", Proceedings of 2007 Information Workshop on Signal Design and Its Applications In Communication, pp.409-412, 2007.
[18]	J.Zheng, T.Kaida, “A note on the shift bound for cyclic codes by the DFT", IEICE Transaction of Fundamentals, Vol.E93-A, No.11, pp.1918-1921, 2010.
[19]	J.Zheng, T.Kaida, “An algorithm for new lower bound of minimum distance by DFT for cyclic codes", Proceedings of 2010 International Symposium on Information Theory and its Applications, pp.846-849, 2010.
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APA Style

Takayasu Kaida, Junru Zheng. (2015). A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes. Pure and Applied Mathematics Journal, 4(2-1), 36-41. https://doi.org/10.11648/j.pamj.s.2015040201.17

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ACS Style

Takayasu Kaida; Junru Zheng. A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes. Pure Appl. Math. J. 2015, 4(2-1), 36-41. doi: 10.11648/j.pamj.s.2015040201.17

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AMA Style

Takayasu Kaida, Junru Zheng. A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes. Pure Appl Math J. 2015;4(2-1):36-41. doi: 10.11648/j.pamj.s.2015040201.17

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@article{10.11648/j.pamj.s.2015040201.17,
  author = {Takayasu Kaida and Junru Zheng},
  title = {A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {2-1},
  pages = {36-41},
  doi = {10.11648/j.pamj.s.2015040201.17},
  url = {https://doi.org/10.11648/j.pamj.s.2015040201.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040201.17},
  abstract = {The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.},
 year = {2015}
}

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TY  - JOUR
T1  - A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes
AU  - Takayasu Kaida
AU  - Junru Zheng
Y1  - 2015/02/11
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.s.2015040201.17
DO  - 10.11648/j.pamj.s.2015040201.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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EP  - 41
PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.pamj.s.2015040201.17
AB  - The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.
VL  - 4
IS  - 2-1
ER  -

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Author Information

Takayasu Kaida

Department Information and Computer Sciences, Faculty of Humanity-Oriented Science and Engineering, Kinki University, Iizuka, Fukuoka, Japan
Junru Zheng

Department of Human Development, Faculty of Humanities, Kyushu Women’s University, Kitakyushu, Fukuoka, Japan

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APA Style

Takayasu Kaida, Junru Zheng. (2015). A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes. Pure and Applied Mathematics Journal, 4(2-1), 36-41. https://doi.org/10.11648/j.pamj.s.2015040201.17

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ACS Style

Takayasu Kaida; Junru Zheng. A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes. Pure Appl. Math. J. 2015, 4(2-1), 36-41. doi: 10.11648/j.pamj.s.2015040201.17

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AMA Style

Takayasu Kaida, Junru Zheng. A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes. Pure Appl Math J. 2015;4(2-1):36-41. doi: 10.11648/j.pamj.s.2015040201.17

Copy | Download

@article{10.11648/j.pamj.s.2015040201.17,
  author = {Takayasu Kaida and Junru Zheng},
  title = {A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {2-1},
  pages = {36-41},
  doi = {10.11648/j.pamj.s.2015040201.17},
  url = {https://doi.org/10.11648/j.pamj.s.2015040201.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040201.17},
  abstract = {The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes
AU  - Takayasu Kaida
AU  - Junru Zheng
Y1  - 2015/02/11
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.s.2015040201.17
DO  - 10.11648/j.pamj.s.2015040201.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 36
EP  - 41
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2015040201.17
AB  - The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.
VL  - 4
IS  - 2-1
ER  -

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