A Method for Constructing Parity-Check Matrices of Quasi-Cyclic LDPC Codes Over GF(q)


An algorithm for constructing parity-check matrices of non-binary quasi-cyclic low-density parity-check (NB QC-LDPC) codes is proposed. The algorithm finds short cycles in the base matrix and tries to eliminate them by selecting the circulants and the elements of GF(q). The algorithm tries to eliminate the cycles with the smallest number edges going outside the cycle. The efficiency of the algorithm is demonstrated by means of simulations. In order to explain the simulation results we also derive upper bounds on the minimum distance of NB QC-LDPC codes.

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    The weight of a circulant is a weight of its first row.


  1. 1

    R. M. Tanner, “On quasi-cyclic repeat-accumulate codes,” in Proc. 37th Allerton Conf. Commun., Contr., Comput., Monticello, IL, Sept. 22–24, 1999 (Allerton House, 1999), pp. 249–259.

  2. 2

    M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes from circulant permutation matrices,” IEEE Trans. Inf. Theory 50, 1788–1793 (2004).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    R. G. Gallager, Low-Density Parity-Check Codes (M.I.T. Press, Cambridge, MA, 1963).

    Google Scholar 

  4. 4

    R. M. Tanner, “A recursive approach to low-complexity codes,” IEEE Trans. Inf. Theory 27, 533–547 (1981).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    J. Thorpe, “Low-density parity-check (LDPC) codes constructed from protographs,” JPL, IPN Progress Rep. 42, 154 (2003).

    Google Scholar 

  6. 6

    Z. Li, L. Chen, L. Zeng, S. Lin, and W. H. Fong, “Efficient encoding of quasi-cyclic low-density parity-check codes,” IEEE Trans. Commun. 54, 71–78 (2006).

    Article  Google Scholar 

  7. 7

    F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory 47, 498–519 (2001).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    M. Davey and D. MacKay, “Low-density parity check codes over GF(q),” IEEE Commun. Lett. 2, 165–167 (1998).

    Article  Google Scholar 

  9. 9

    H. Song and J. R. Cruz, “Reduced-complexity decoding of Q-ary LDPC codes for magnetic recording,” IEEE Trans. Magnetics 39, 1081–1087 (2003).

    Article  Google Scholar 

  10. 10

    H. Wymeersch, H. Steendam, and M. Moeneclaey, “Log-domain decoding of LDPC codes over GF(q),” in Proc. IEEE Int. Conf. on Communications,Paris, June 20–24, 2004 (IEEE, Piscataway, 2004), pp. 772–776.

  11. 11

    L. Barnault and D. Declercq, “Fast decoding algorithm for LDPC over GF(2q),” in Proc. 2003 Inf. Theory Workshop, Paris, France, Mar. 2003 (IEEE, Piscataway, 2003), pp. 70–73.

  12. 12

    D. Declercq and M. Fossorier, “Decoding algorithms for aonbinary LDPC codes over GF(q),” IEEE Trans. on Commun. 55, 633–643 (2007).

    Article  Google Scholar 

  13. 13

    G. Liva and M. Chiani, “Protograph LDPC codes design based on EXIT analysis,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM 2007), Washington, DC, Nov. 26–30, 2007 (IEEE, New York, 2007), pp. 3250–3254.

  14. 14

    T. Y. Chen, K. Vakilinia, D. Divsalar, and R. D. Wesel, “Protograph-based raptor-like LDPC codes,” IEEE Trans. Commun. 63, 1522–1532 (2015).

    Article  Google Scholar 

  15. 15

    A. Bazarsky, N. Presman, and S. Litsyn, “Design of non-binary quasi-cyclic LDPC codes by ACE optimization,” in Proc. IEEE Inf. Theory Workshop, Sevilla, Spain, 2013 (IEEE, New York, 2013).

  16. 16

    H. Xiao and A. H. Banihashemi, “Improved progressive-edge-growth (PEG) construction of irregular LDPC codes,” IEEE Commun. Lett. 8, 715–717 (2004).

    Article  Google Scholar 

  17. 17

    X. Hu, E. Eleftheriou, and D. Arnold, “Irregular progressive edge-growth (PEG) tanner graphs,” in Proc. ISIT 2002, Lausanne, Switzerland, June 30–July 5, 2002 (IEEE, New York, 2002).

  18. 18

    T. Richardson and R. Urbanke, Modern Coding Theory (Cambridge Univ. Press, Cambridge, 2008).

    Google Scholar 

  19. 19

    R. Smarandache and P. O. Vontobel, “Quasi-cyclic LDPC codes: influence of proto- and tanner-graph structure on minimum hamming distance upper bounds,” IEEE Trans. Inf. Theory 58, 585–607 (2012).

    MathSciNet  Article  MATH  Google Scholar 

  20. 20

    D. J. C. MacKay and M. C. Davey, “Evaluation of Gallager codes for short block length and high rate applications,” in Codes, Systems, and Graphical Models, Minneapolis, MN, 1999, Ed. by B. Marcus and J. Rosenthal (Springer-Verlag, New York, 2001), pp. 113–130.

    Google Scholar 

  21. 21

    T. Richardson and R. Urbanke, Modern Coding Theory (Cambridge Univ. Press, Cambridge, U. K., 2008).

    Google Scholar 

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The research was carried out at Skoltech and supported by the Russian Science Foundation (project no. 18-19-00673).

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Correspondence to S. A. Kruglik or V. S. Potapova or A. A. Frolov.

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The article was translated by the authors.

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Kruglik, S.A., Potapova, V.S. & Frolov, A.A. A Method for Constructing Parity-Check Matrices of Quasi-Cyclic LDPC Codes Over GF(q). J. Commun. Technol. Electron. 63, 1524–1529 (2018). https://doi.org/10.1134/S1064226918120112

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  • LDPC code
  • parity-check matrix
  • iterative decoding threshold
  • Tanner graph
  • cycle
  • Galois field