Advertisement

Journal of Global Optimization

, Volume 55, Issue 2, pp 301–312 | Cite as

Monotonic optimization based decoding for linear codes

  • H. D. TuanEmail author
  • T. T. Son
  • H. Tuy
  • P. T. Khoa
Article

Abstract

New efficient methods are developed for the optimal maximum-likelihood (ML) decoding of an arbitrary binary linear code based on data received from any discrete Gaussian channel. The decoding algorithm is based on monotonic optimization that is minimizing a difference of monotonic (d.m.) objective functions subject to the 0–1 constraints of bit variables. The iterative process converges to the global optimal ML solution after finitely many steps. The proposed algorithm’s computational complexity depends on input sequence length k which is much less than the codeword length n, especially for a codes with small code rate. The viability of the developed is verified through simulations on different coding schemes.

Keywords

Linear codes Low density parity check (LDPC) codes Maximum likelihood decoding Global optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berlekamp E.R., McElice R.J., van Tiborg H.C.A.: On the intractibility of certain coding problems. IEEE Trans. Inf. Theory 24, 384–386 (1978)CrossRefGoogle Scholar
  2. 2.
    Feldman J., Wainwright M.J., Karger D.R.: Using linear programming to decode binary linear codes. IEEE Trans. Inf. Theory 51, 954–972 (2005)CrossRefGoogle Scholar
  3. 3.
    Frank M., Wolfe P.: An algorithm for quadratic programming. Naval Res. Log. Q. 3, 95–110 (1956)CrossRefGoogle Scholar
  4. 4.
    Gallager R.G.: Low Density Parity Check Codes. MIT Press, Cambridge (1962)Google Scholar
  5. 5.
    Hasselberg J., Pardalos P.M., Vairaktarakis G.: Test case generators and computational results for the maximum clique problem. J. Glob. Optim. 3, 463–482 (1993)CrossRefGoogle Scholar
  6. 6.
    Kschischang F.R., Frey B.J., Loeliger H.A.: Factor graphs and sum-product algorithm. IEEE Trans. Inf. Theory 47, 498–519 (2001)CrossRefGoogle Scholar
  7. 7.
    LDPC toolkit for Matlab. http://arun-10.tripod.com/ldpc/ldpc.htm
  8. 8.
    Mackay D.J.C., Neal R.M.: Near Shannon limit performance of low density parity check codes. Electron. Lett. 32, 1645–1646 (1996)CrossRefGoogle Scholar
  9. 9.
    Mackay D.J.C.: Good error-correcting codes based on very sparse matrices. IEEE Trans. Inf. Theory 45, 399–431 (1999)CrossRefGoogle Scholar
  10. 10.
    Margulis G.A.: Explicit constructions of graphs without short cycles and low density codes. Combinatorica 2, 71–78 (1982)CrossRefGoogle Scholar
  11. 11.
    McEliece R., MacKay D., Cheng J.: Turbo decoding as an instance of Pearl’s belief propagation algorithm. IEEE J. Sel. Areas Commun. 16, 140–152 (1998)CrossRefGoogle Scholar
  12. 12.
    Pardalos P.M., Romeijn E., Tuy H.: Recent developments and trends in global optimization. J. Comput. Appl. Math. 124, 209–228 (2000)CrossRefGoogle Scholar
  13. 13.
    Pearl J.: Probabilistic Reasoning in Intelligent Systems. Margan Kaufmann, Los Altos (1988)Google Scholar
  14. 14.
    Tanner R.M.: A recursive approach to low complexity codes. IEEE Trans. Inf. Theory 27, 533–547 (1981)CrossRefGoogle Scholar
  15. 15.
    Tuy H.: Convex Analysis and Global Optimization. Kluwer Academic, New York (1999)Google Scholar
  16. 16.
    Tuy H., Minoux M., Phuong N.T.H.: Discrete monotonic optimization with application to a discrete location problem. SIAM J. Optim. 17, 78–97 (2006)CrossRefGoogle Scholar
  17. 17.
    Tuy H.: Monotonic optimization: problems and solution approaches. SIAM J. Optim. 11, 464–494 (2000)CrossRefGoogle Scholar
  18. 18.
    Yang K., Feldman J., Wang X.: Nonlinear progrramming approachs to decoding low-density parity-check codes. IEEE J. Sel. Areas Commun. 24, 1603–1613 (2006)CrossRefGoogle Scholar
  19. 19.
    Yedidia J.S., Freeman W.T., Weiss Y.: Understanding Belief Propagation and its Generalizations. In: Lakemeyer, G., Nebel, B. (eds) Exploring Artificial Intelligence in the New Millenium chap 8., pp. 239–269. Morgan Kaufmann, CA (2003)Google Scholar
  20. 20.
    Yedidia J.S., Freeman W.T., Weiss Y.: Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. Inf. Theory 51, 2282–2312 (2005)CrossRefGoogle Scholar
  21. 21.
    Yuille A.L.: CCCP algorithms to minimize the Bethe and Kikuchi energies: convergent alternatives to belief propagation. Neural Comput. 14, 1691–1722 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Faculty of Engineering and Information TechnologyUniversity of TechnologySydneyAustralia
  2. 2.Department of Computer Vision and Robotics-FITUniversity of ScienceHochiminh CityVietnam
  3. 3.Institute of MathematicsHanoiVietnam
  4. 4.Department of Electrical EngineeringUniversity of California in Los Angeles (UCLA)Los AngelesUSA

Personalised recommendations