Skip to main content
SpringerLink
Log in

Fast Decoding Algorithms for First Order Reed-Muller and Related Codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Fast decoding algorithms for short codes based on modifications of maximum likelihood decoding algorithms of first order Reed-Muller codes are described. Only additions-subtractions, comparisons and absolute value calculations are used in the algorithms. Soft and hard decisions maximum likelihood decoding algorithms for first order Reed-Muller and the Nordstrom-Robinson codes with low complexity are proposed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J-P Adoul, Fast ML decoding algorithm for the Nordstrom-Robinson code, IEEE Trans. nform. Theory Vol. IT-33 (1987) pp. 931–933.

    Google Scholar 

  2. A. Ashikhmin. A. Dmitriyov, S. Litsyn and S. Portnoy, Patent USSR, No 1133980 (1988).

  3. A. Ashikhmin and S. Litsyn, Analysis of quasi-optimal decoding algorithms of biorthogonal codes, Radio-electronica Vol31, No. 11 (1988) pp. 30–34( in Russian).

    Google Scholar 

  4. A. Ashikhmin and S. Litsyn, Fast decoding algorithms for first order Reed-Muller and related codes, Tel-Aviv University, Reponrt EE93–2 (1993).

  5. A. Ashikhmin and S. Litsyn, List algorithm for search of the maximal element of Walsh transform, Radio-electronica Vol 32, No. 3 (1990) pp 15–22(In Russian).

    Google Scholar 

  6. Y. Be'ery and J. Snyders, Optimal soft decision block decoders based on fast Hadamard transform, IEEE Trans. Inform. Theory Vol. IT32 (1986) pp. 355–364.

    Google Scholar 

  7. E. R. Berlekamp, The technology of error-conecting codes, Proc. IEEE, Vol68 (1980) pp.355–364

    Google Scholar 

  8. E. Berlekamp and L. R. Welch, Weight distributions of the cosets of the (32,6) Reed-Muller code. IEEE Trans Inform. Theory. Vol. IT-18 (1972) pp.203–207.

    Google Scholar 

  9. J. H. Conway and N J. A. Sloane, Fast quantizing and decoding algorithms for lattice quantizers and codes, IEEE Trans. Inform. Theory Vol. IT-28 (1982) pp. 272–232.

    Google Scholar 

  10. J. H. Conway, N.J. A. Sloane, Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice, IEEE Trans. Inform Theory. Vol. [vnIT 32], No. I1 (1986) pp. 41–50.

  11. J H. Conway and N. J A. Sloane, Sphere Packings, Lattices andGroups Springer-Verlag, New York (1988).

    Google Scholar 

  12. G. D. Fomey,Jr., Coset codes II: Binary lattices and related codes, IEEE Trans Inform. Theory. Vol. IT-34 (1988)pp. 1152–1187.

    Google Scholar 

  13. S.W. Golomb, ed., Digital Commumncatons with Space Applicatons. Prentice-Hall, Englewood CliffS NJ (1964).

    Google Scholar 

  14. R R. Green, A serial orthogonal decoder, JPL Space Programs Summary Vol37 39 IV (1966) pp. 247–253.

    Google Scholar 

  15. C. M. Hackett, An efficient algorithm for soft decision decoding of the (24,12) extended Golay code, IEEE Trans. Commun., Vol.Com-29 (1981) pp. 909 911, and Vol. Com 30 (1982) p. 554.

    Google Scholar 

  16. S. Litsyn, G. Mikhailovskaya, E. Nemirovsky and 0. Shekhovtsov, Fast decoding of first order Reed-Mnller indes in the Galssian channel, Prolemr of Control and Information Theory. Vol14. No. 3 (1985) pp. 189–201.

    Google Scholar 

  17. S. Litsyn and 0. Shekhovtsov, Fast decoding algorithm for first order Reed-Muller codes, Problems of Information Transmission Vol19, No.2 (1983) pp. 3–7.

    Google Scholar 

  18. J. A. Maiorna, A classification of the sets of the Reed Muller code 72(16) Mathcmatics of Computation Vol57, No. 195 (1991) pp. 403–414.

    Google Scholar 

  19. E J. MacWilliams and N. J. A. Sloane, The Theory of Errorcorecing Codes North-Holland, Amsterdam, The Netherlands (1977).

    Google Scholar 

  20. H. J. Manley, H. F. Mattson and J. R. Schatz, Some applications of Good's theorem, IEEE Trans. nform. Theory Vol. IT-26 (1980) pp. 475–476.

    Google Scholar 

  21. V. Pless, Decoding the Golay codes, IEEE Trans. Inform. Theory, Vol. IT-32 (1986) pp. 561 567.

    Google Scholar 

  22. N. J. A. Sloane and R. J. Dick, On the enumeration of cosets of first order Reed-Muller codes, Proc. IEEE Int Conf Communications Vol. 7 (1971) 36–2–36–6.

    Google Scholar 

  23. J. Snyders and Y. Be'ery, Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes, IEEE Trans. Inform. Theory Vol. IT-35 (1986) pp. 963–975.

    Google Scholar 

  24. J. Wolfmann, A permutation decoding of the (24,12,8) Golay code, IEEE Trans. Inform. Theory, Vol.IT-29 (1983) pp. 748–750.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Problems of Information Transmission, Ermolovoy 19, Moscow, Russia

    Alexey E. Ashikhmin

  2. Department of Electrical Engineering-Systems, Tel Aviv University, Ramat Aviv, 69978, Israel

    Simon N. Litsyn

Authors
  1. Alexey E. Ashikhmin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon N. Litsyn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ashikhmin, A.E., Litsyn, S.N. Fast Decoding Algorithms for First Order Reed-Muller and Related Codes. Designs, Codes and Cryptography 7, 187–214 (1996). https://doi.org/10.1023/A:1018057506207

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018057506207

Search

Navigation