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Algebraic Coding 1993: Algebraic Coding pp 236-248 | Cite as

Bounded-distance decoding of the Leech lattice and the Golay code

  • Ofer Amrani
  • Yair Be'ery
  • Alexander Vardy
Sphere Packings and Lattices
Part of the Lecture Notes in Computer Science book series (LNCS, volume 781)

Abstract

We present an efficient algorithm for bounded-distance decoding of the Leech lattice. The new bounded-distance algorithm employs the partition of the Leech lattice into four cosets of Q24 beyond the conventional partition into two H24 cosets. The complexity of the resulting decoder is only 1007 real operations in the worst case, as compared to about 3600 operations for the best known maximum-likelihood decoder and about 2000 operations for the original bounded-distance decoder of Forney. Restricting the proposed Leech lattice decoder to GF(2)24 yields a bounded-distance decoder for the binary Golay code which requires at most 431 operations as compared to 651 operations for the best known maximum-likelihood decoder. Moreover, it is shown that our algorithm decodes correctly at least up to the guaranteed error-correction radius of the Leech lattice. Performance of the algorithm on the AWGN channel is evaluated analytically by explicitly calculating the effective error-coefficient, and experimentally by means of a comprehensive computer simulation. The results show a loss in coding-gain of less than 0.1 dB relative to the maximum-likelihood decoder for BER ranging from 10−1 to 10−7.

Keywords

Additive White Gaussian Noise Channel Real Operation AWGN Channel Comprehensive Computer Simulation Golay Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E.F. Assmus, Jr. and H.F. Mattson, Jr., Algebraic Theory of Codes, Report No.1, Contract No. F19628-69C0068, Air Force Cambridge Research Labs., Bedford, MA., 1969.Google Scholar
  2. 2.
    Y. Be'ery, B. Shahar, and J. Snyders, “Fast decoding of the Leech lattice,” IEEE J. Select. Areas Comm., vol. SAC-7, pp. 959–967, 1989.Google Scholar
  3. 3.
    J.H. Conway and N.J.A. Sloane, “Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 41–50, 1986.Google Scholar
  4. 4.
    J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, New York: Springer-Veralg, 1988.Google Scholar
  5. 5.
    G.D. Forney, Jr., “Coset codes II: Binary lattices and related codes,” IEEE Trans. Inform. Theory, vol. IT-34, pp. 1152–1187, 1988.Google Scholar
  6. 6.
    G.D. Forney, Jr., “A bounded distance decoding algorithm for the Leech lattice, with generalizations,” IEEE Trans. Inform. Theory, vol. IT-35, pp. 906–909, 1989.Google Scholar
  7. 7.
    G.R. Lang and F.M. Longstaff, “A Leech lattice modem,” IEEE J. Select. Areas Comm., vol. SAC-7, pp. 968–973, 1989.Google Scholar
  8. 8.
    V. Pless and N.J.A. Sloane, “On the classification of self dual codes,” J. Comb. Theory, vol. 18A, pp. 313–335, 1975.Google Scholar
  9. 9.
    M. Ran and J. Snyders, “On maximum likelihood soft decoding of some binary self dual codes,” to appear in IEEE Trans. Comm., 1993.Google Scholar
  10. 10.
    R. A. Silverman and M. Balser, “Coding for a constant data rate source,” IRE Trans. Inform. Theory, vol. 4, pp. 50–63, 1954.Google Scholar
  11. 11.
    A. Vardy and Y. Be'ery, “More efficient soft-decision decoding of the Golay codes,” IEEE Trans. Inform. Theory, vol. IT-37, pp. 667–672, 1991.Google Scholar
  12. 12.
    A. Vardy and Y. Be'ery, “Maximum-likelihood decoding of the Leech lattice,” to appear in IEEE Trans. Inform. Theory, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Ofer Amrani
    • 1
  • Yair Be'ery
    • 1
  • Alexander Vardy
    • 2
  1. 1.Department of Electrical Engineering - SystemsTel-Aviv UniversityTel-AvivIsrael
  2. 2.IBM Research DivisionAlmaden Research CenterSan Jose

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