A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26

Li Peng

doi:doi:10.11648/j.com.20140203.11

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A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26

Li Peng

Published in Communications (Volume 2, Issue 3)

Received: 14 December 2014 Accepted: 28 December 2014 Published: 14 January 2015

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Abstract

This paper presents a method of constructing the half-rate irregular quasi-cyclic low-density parity-check codes which can provide linear encoding algorithm and their H-matrices may contain almost the least “1” elements comparing with H-matrices of all existing LDPC codes. This method shows that three kinds of special structural matrices, respectively named as S-matrix, M-matrix and A-matrix, are defined and constructed. With regard to the arbitrary large structural girth based on A-matrix, its general pattern is conceived and its basic rule is proved. A general method of constructing M-matrix with the inevitable girth larger than 24 is introduced by using generalized block design and treating A-matrix as its sub-matrix. S-matrix is generated by substituting specially circular-shift values for non-zero elements in M-matrix. Combining Hd-matrix generated from lifting the S-matrix and Hp-matrix with the approximate lower triangular array structure forms the H-matrix, i.e. H=[Hd Hp], which defines a class of half-rate irregular QC-LDPC codes with maximum column weight 3 and inevitable girth of length 26. Simulation tests show that the performance of the presented QC-LDPC code can achieve the signal-noise-ratio of below 2dB at the bit-error-rate of 10-5, which is comparable with the performance of the practical QC-LDPC codes in industrial Standard, but the complication of the former owing to the least “1” elements in H-matrix is lower than that of the later, as well as the storage requirement is smaller.

Published in	Communications (Volume 2, Issue 3)
DOI	10.11648/j.com.20140203.11
Page(s)	22-34
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), 2024. Published by Science Publishing Group

Keywords

Quasi-Cycle Low-Density Parity-Check (QC-LDPC) Code, Sparse Parity-Check Matrix, Girth, Generalized Block Design

References

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[3]	Gallager R. G., Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1963.
[4]	J. L. Fan, “Array Codes as Low-Density Parity-Check Codes”, in Proc. 2nd Int. Symp. Turbo Codes and Related Topics, Brest, France, Sept. 2000, pp. 543-546.
[5]	R. M. Tanner, D. Sridhara, A. Sridharan, T. E. Fuja and D. J. Costello, “ LDPC Block and Convolutional Codes Based on Circular Matrices”, IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 2966-2984, Dec. 2004.
[6]	M. P. Fossorier, “Quasi-Cyclic Low-Density Parity- Check Codes From Circular Permutation Matrices”, IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1788-1793, Aug. 2004.
[7]	Zongwang Li, Lei Chen, Lingqi Zeng, Shu Lin and Wai H. Fong, “Efﬁcient Encoding of Quasi-Cyclic Low- Density Parity-Check Codes”, IEEE Trans. on Commu., vol. 54, no. 1, pp. 71-81, Jan. 2006.
[8]	Olgica Milenkovic, Navin Kashyap and David Leyba, “Shortened Array Codes of Large Girth”, IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3707-3723, Agu. 2006.
[9]	Sunghwan Kim, Habong Chung and Dong-Joon Shin, “Quasi-Cyclic Low-Density Parity-Check Codes With Girth Larger Than 12”, IEEE Trans. Inf. Theory, vol. 53, no. 8, pp. 2885-2891, Aug. 2007.
[10]	R. A. Brualdi, “Introductory Combinatorics,” Third Edition, Prentice Hall, 1999.
[11]	Lan Lan, L. Q. Zeng, Y. Y. Tai, L. Chen, S. Lin and K. A-Ghaffar, “Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach,” IEEE Trans. Inf. Theory, vol. 53, no. 7, pp. 2429-2458, July 2007.
[12]	M. Gholami, M. Samadieh and G. Raeisi, “Column-Weight Three QC LDPC Codes with Girth 20,” IEEE Commun. Lett., vol. 17, no. 7, pp. 1439–1442, July 2013.
[13]	M. Gholami and G. Raeisi, “Large Girth Column- Weight Two and Three LDPC Codes,” IEEE Commun. Lett., vol. 18, no. 10, pp. 1671–1674, Oct. 2014.

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APA Style

Li Peng. (2015). A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26. Communications, 2(3), 22-34. https://doi.org/10.11648/j.com.20140203.11

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Li Peng. A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26. Communications. 2015, 2(3), 22-34. doi: 10.11648/j.com.20140203.11

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AMA Style

Li Peng. A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26. Communications. 2015;2(3):22-34. doi: 10.11648/j.com.20140203.11

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@article{10.11648/j.com.20140203.11,
  author = {Li Peng},
  title = {A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26},
  journal = {Communications},
  volume = {2},
  number = {3},
  pages = {22-34},
  doi = {10.11648/j.com.20140203.11},
  url = {https://doi.org/10.11648/j.com.20140203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.com.20140203.11},
  abstract = {This paper presents a method of constructing the half-rate irregular quasi-cyclic low-density parity-check codes which can provide linear encoding algorithm and their H-matrices may contain almost the least “1” elements comparing with H-matrices of all existing LDPC codes. This method shows that three kinds of special structural matrices, respectively named as S-matrix, M-matrix and A-matrix, are defined and constructed. With regard to the arbitrary large structural girth based on A-matrix, its general pattern is conceived and its basic rule is proved. A general method of constructing M-matrix with the inevitable girth larger than 24 is introduced by using generalized block design and treating A-matrix as its sub-matrix. S-matrix is generated by substituting specially circular-shift values for non-zero elements in M-matrix. Combining Hd-matrix generated from lifting the S-matrix and Hp-matrix with the approximate lower triangular array structure forms the H-matrix, i.e. H=[Hd Hp], which defines a class of half-rate irregular QC-LDPC codes with maximum column weight 3 and inevitable girth of length 26. Simulation tests show that the performance of the presented QC-LDPC code can achieve the signal-noise-ratio of below 2dB at the bit-error-rate of 10-5, which is comparable with the performance of the practical QC-LDPC codes in industrial Standard, but the complication of the former owing to the least “1” elements in H-matrix is lower than that of the later, as well as the storage requirement is smaller.},
 year = {2015}
}

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TY  - JOUR
T1  - A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26
AU  - Li Peng
Y1  - 2015/01/14
PY  - 2015
N1  - https://doi.org/10.11648/j.com.20140203.11
DO  - 10.11648/j.com.20140203.11
T2  - Communications
JF  - Communications
JO  - Communications
SP  - 22
EP  - 34
PB  - Science Publishing Group
SN  - 2328-5923
UR  - https://doi.org/10.11648/j.com.20140203.11
AB  - This paper presents a method of constructing the half-rate irregular quasi-cyclic low-density parity-check codes which can provide linear encoding algorithm and their H-matrices may contain almost the least “1” elements comparing with H-matrices of all existing LDPC codes. This method shows that three kinds of special structural matrices, respectively named as S-matrix, M-matrix and A-matrix, are defined and constructed. With regard to the arbitrary large structural girth based on A-matrix, its general pattern is conceived and its basic rule is proved. A general method of constructing M-matrix with the inevitable girth larger than 24 is introduced by using generalized block design and treating A-matrix as its sub-matrix. S-matrix is generated by substituting specially circular-shift values for non-zero elements in M-matrix. Combining Hd-matrix generated from lifting the S-matrix and Hp-matrix with the approximate lower triangular array structure forms the H-matrix, i.e. H=[Hd Hp], which defines a class of half-rate irregular QC-LDPC codes with maximum column weight 3 and inevitable girth of length 26. Simulation tests show that the performance of the presented QC-LDPC code can achieve the signal-noise-ratio of below 2dB at the bit-error-rate of 10-5, which is comparable with the performance of the practical QC-LDPC codes in industrial Standard, but the complication of the former owing to the least “1” elements in H-matrix is lower than that of the later, as well as the storage requirement is smaller.
VL  - 2
IS  - 3
ER  -

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Author Information

Li Peng

Wuhan National Laboratory for Optoelectronics, School of Electronic Information and Communications in Huazhong University of Science and Technology, Wuhan, China, 430074

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APA Style

Li Peng. (2015). A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26. Communications, 2(3), 22-34. https://doi.org/10.11648/j.com.20140203.11

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ACS Style

Li Peng. A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26. Communications. 2015, 2(3), 22-34. doi: 10.11648/j.com.20140203.11

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AMA Style

Li Peng. A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26. Communications. 2015;2(3):22-34. doi: 10.11648/j.com.20140203.11

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@article{10.11648/j.com.20140203.11,
  author = {Li Peng},
  title = {A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26},
  journal = {Communications},
  volume = {2},
  number = {3},
  pages = {22-34},
  doi = {10.11648/j.com.20140203.11},
  url = {https://doi.org/10.11648/j.com.20140203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.com.20140203.11},
  abstract = {This paper presents a method of constructing the half-rate irregular quasi-cyclic low-density parity-check codes which can provide linear encoding algorithm and their H-matrices may contain almost the least “1” elements comparing with H-matrices of all existing LDPC codes. This method shows that three kinds of special structural matrices, respectively named as S-matrix, M-matrix and A-matrix, are defined and constructed. With regard to the arbitrary large structural girth based on A-matrix, its general pattern is conceived and its basic rule is proved. A general method of constructing M-matrix with the inevitable girth larger than 24 is introduced by using generalized block design and treating A-matrix as its sub-matrix. S-matrix is generated by substituting specially circular-shift values for non-zero elements in M-matrix. Combining Hd-matrix generated from lifting the S-matrix and Hp-matrix with the approximate lower triangular array structure forms the H-matrix, i.e. H=[Hd Hp], which defines a class of half-rate irregular QC-LDPC codes with maximum column weight 3 and inevitable girth of length 26. Simulation tests show that the performance of the presented QC-LDPC code can achieve the signal-noise-ratio of below 2dB at the bit-error-rate of 10-5, which is comparable with the performance of the practical QC-LDPC codes in industrial Standard, but the complication of the former owing to the least “1” elements in H-matrix is lower than that of the later, as well as the storage requirement is smaller.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - A Method of Constructing the Half-Rate QC-LDPC Codes with Linear Encoder, Maximum Column Weight Three and Inevitable Girth 26
AU  - Li Peng
Y1  - 2015/01/14
PY  - 2015
N1  - https://doi.org/10.11648/j.com.20140203.11
DO  - 10.11648/j.com.20140203.11
T2  - Communications
JF  - Communications
JO  - Communications
SP  - 22
EP  - 34
PB  - Science Publishing Group
SN  - 2328-5923
UR  - https://doi.org/10.11648/j.com.20140203.11
AB  - This paper presents a method of constructing the half-rate irregular quasi-cyclic low-density parity-check codes which can provide linear encoding algorithm and their H-matrices may contain almost the least “1” elements comparing with H-matrices of all existing LDPC codes. This method shows that three kinds of special structural matrices, respectively named as S-matrix, M-matrix and A-matrix, are defined and constructed. With regard to the arbitrary large structural girth based on A-matrix, its general pattern is conceived and its basic rule is proved. A general method of constructing M-matrix with the inevitable girth larger than 24 is introduced by using generalized block design and treating A-matrix as its sub-matrix. S-matrix is generated by substituting specially circular-shift values for non-zero elements in M-matrix. Combining Hd-matrix generated from lifting the S-matrix and Hp-matrix with the approximate lower triangular array structure forms the H-matrix, i.e. H=[Hd Hp], which defines a class of half-rate irregular QC-LDPC codes with maximum column weight 3 and inevitable girth of length 26. Simulation tests show that the performance of the presented QC-LDPC code can achieve the signal-noise-ratio of below 2dB at the bit-error-rate of 10-5, which is comparable with the performance of the practical QC-LDPC codes in industrial Standard, but the complication of the former owing to the least “1” elements in H-matrix is lower than that of the later, as well as the storage requirement is smaller.
VL  - 2
IS  - 3
ER  -

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