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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

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), 2015. Published by Science Publishing Group

Keywords

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

References
[1] Part 16: Air Interface for Fixed and Mobile Broadband wireless Access Systems, IEEE Standard 802.16e, 2006.
[2] 2. Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications, IEEE Standard 802.11n, 2007.
[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.
Cite This Article
  • 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}
    }
    

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    AU  - Li Peng
    Y1  - 2015/01/14
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    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
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Author Information
  • 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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