Skip to main content
SpringerLink
Log in

Trellis Structure and Higher Weights of Extremal Self-Dual Codes

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

Abstract

A method for demonstrating and enumerating uniformly efficient (permutation-optimal) trellis decoders for self-dual codes of high minimum distance is developed. Such decoders and corresponding permutations are known for relatively few codes.

The task of finding such permutations is shown to be substantially simplifiable in the case of self-dual codes in general, and for self-dual codes of sufficiently high minimum distance it is shown that it is frequently possible to deduce the existence of these permutations directly from the parameters of the code.

A new and tighter link between generalized Hamming weights and trellis representations is demonstrated: for some self-dual codes, knowledge of one of the generalized Hamming weights is sufficient to determine the entire optimal state complexity profile.

These results are used to characterize the permutation-optimal trellises and generalized Hamming weights for all [32,16,8] binary self-dual codes and for several other codes. The numbers of uniformly efficient permutations for several codes, including the [24,12,8] Golay code and both [24,12,9] ternary self-dual codes, are found.

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. L. R. Bahl, J. Cocke, F. Jelinek and J. Raviv, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans. Inform. Theory, Vol. IT–20, No. 2 (1974) pp. 284–287.

    Google Scholar 

  2. Y. Berger and Y. Be'ery, The twisted squaring construction, trellis complexity and generalized weights of BCH and QR codes, IEEE Trans. Inform. Theory, Vol. IT–42, No. 6 (1996) pp. 1817–1827.

    Google Scholar 

  3. A. E. Brouwer and T. Verhoeff, An updated table of minimum-distance bounds for binary linear codes, IEEE Trans. Inform. Theory, Vol. 39, No. 2 (1993) pp. 662–677.

    Google Scholar 

  4. H. Chen, Optimal encoding, trellis structure, and normalized weight of linear block codes, PhD. Dissertation, University of Michigan, May (1999).

  5. J. H. Conway and V. Pless, On the enumeration of self-dual codes, J. Combin. Theory Ser. A , Vol. 28, No. 1 (1980) pp. 26–53.

    Google Scholar 

  6. J. H. Conway, V. Pless, and N. J. A. Sloane, Self-dual codes over GF(3) and GF(4) of length not exceeding 16, IEEE Trans. Inform. Theory, Vol. IT-25 (1979) pp. 312–322.

    Google Scholar 

  7. J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, Vol. 36, No. 6 (1990) pp. 1319–1333.

    Google Scholar 

  8. J. H. Conway and N. J. A. Sloane, Orbit and coset analysis of the Golay and related codes, IEEE Trans. Inform. Theory, Vol. IT-36, No. 5 (1990) pp. 1038–1050.

    Google Scholar 

  9. S. M. Dodunekov and S. B. Encheva, Uniqueness of some linear subcodes of the extended binary Golay code, Problemy Peredachi Informatsii, Vol. 29, No. 1 (1993) pp. 45–51. In Russian. English translation in Problems of Information Transmission, Vol. 29 No. 1 (1993) pp. 38–43.

    Google Scholar 

  10. S. Dolinar, L. Ekroot, A. B. Kiely, R. J. McEliece and W. Lin, The permutation trellis complexity of linear block codes, In Proc. 32nd Allerton Conf. on Communications, Control, and Computing, Monticello, IL, pp. 60–74, September (1994).

  11. S. B. Encheva, On the binary linear codes which satisfy the two-way chain condition, IEEE Trans. Inform. Theory, Vol. IT-42, No. 3 (1996) pp. 1038–1047.

    Google Scholar 

  12. S. B. Encheva, On repeated-root cyclic codes and the two-way chain condition, in Lecture Notes Comput. Sci., Vol. 1255, Springer-Verlag (1997) pp. 78–87.

    Google Scholar 

  13. S. B. Encheva and G. D. Cohen, Self-orthogonal codes and their coordinate ordering, IEICE Trans. Fundamentals, Vol. E80-A, No. 11 (1997) pp. 2256–2259.

    Google Scholar 

  14. S. B. Encheva and G. D. Cohen, On the state complexities of ternary codes, In Proceedings of AAECC-13 (Honolulu, Hawaii), November (1999) pp. 454–461.

  15. G. D. Forney Jr., Dimension/length profiles and trellis complexity of linear block codes, IEEE Trans. Inform. Theory, Vol. IT–40, No. 6 (1994) pp. 1741–1752.

    Google Scholar 

  16. T. Helleseth, A characterization of codes meeting the Griesmer bound, Information and Control, Vol. 50,No. 2 (1981) pp. 128–159.

    Google Scholar 

  17. T. Helleseth, T. Kløve and Ø. Ytrehus, Generalized Hamming weights of linear codes, IEEE Trans. Inform. Theory, Vol. IT-38, No. 3 (1992) pp. 1133–1140.

    Google Scholar 

  18. G. B. Horn and F. R. Kschischang, On the intractability of permuting a block code to minimize trellis complexity, IEEE Trans. Inform. Theory, Vol. IT–42, No. 6 (1996) pp. 2042–2048.

    Google Scholar 

  19. T. Kasami, T. Takata, T. Fujiwara and S. Lin, On complexity of trellis structure of linear block codes, IEEE Trans. Inform. Theory, Vol. IT–39, No. 3 (1993) pp. 1057–1064.

    Google Scholar 

  20. T. Kasami, T. Takata, T. Fujiwara and S. Lin, On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes, IEEE Trans. Inform. Theory, Vol. IT–39, No. 1 (1993) pp. 242–245.

    Google Scholar 

  21. A. B. Kiely, S. Dolinar, R. J. McEliece, L. Ekroot and W. Lin, Trellis decoding complexity of linear block codes, IEEE Trans. Inform. Theory, Vol. IT–42, No. 6 (1996) pp. 1687–1697.

    Google Scholar 

  22. H. Koch, On self-dual, doubly-even codes of length 32, J. Combin. Theory Ser. A, Vol. 51, No. 1 (1989) pp. 63–67.

    Google Scholar 

  23. F. R. Kschischang and G. B. Horn, A heuristic for ordering a linear block code to minimize trellis state complexity, In Proc. 32nd Allerton Conf. on Communications, Control, and Computing, Monticello, IL, September (1994) pp. 75–84.

  24. J. S. Leon, V. Pless and N. J. A. Sloane, On ternary self-dual codes of length 24, IEEE Trans. Inform. Theory, Vol. IT-27, No. 2 (1981) pp. 176–180.

    Google Scholar 

  25. S. Lin, T. Kasami, T. Fujiwara and M. Fossorier, Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes, Kluwer Academic, Boston, Massachusetts (1998).

    Google Scholar 

  26. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, The Netherlands (1978).

    Google Scholar 

  27. D. J. Muder, Minimal trellises for block codes, IEEE Trans. Inform. Theory, Vol. IT–34, No. 5 (1988) pp. 1049–1053.

    Google Scholar 

  28. V. S. Pless, Coding constructions, in Handbook of Coding Theory, Volume I, (V. S. Pless and W. C. Huffman, eds.), North-Holland Amsterdam, The Netherlands (1998).

    Google Scholar 

  29. V. S. Pless, W. C. Huffman and R. Brualdi, An introduction to algebraic codes, in Handbook of Coding Theory, Volume I, (V. S. Pless and W. C. Huffman, eds.), North-Holland, Amsterdam, The Netherlands (1998).

    Google Scholar 

  30. E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, Volume I, (V. S. Pless and W. C. Huffman, eds.), North-Holland, Amsterdam, The Netherlands (1998).

    Google Scholar 

  31. J. Simonis, The effective length of subcodes, Appl. Algebra Engrg. Commun. Comput., Vol. 5, No. 6 (1994) pp. 371–377.

    Google Scholar 

  32. M. A. Tsfasman and S. G. Vlăduţ¸, Geometric approach to higher weights, IEEE Trans. Inform. Theory, Vol. 41, No. 6 (1995) pp. 1564–1588.

    Google Scholar 

  33. J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York (1982).

    Google Scholar 

  34. H. C. A. van Tilborg, On the uniqueness resp. nonexistence of certain codes meeting the Griesmer bound, Information and Control, Vol. 44, No. 1 (1980) pp. 16–35.

    Google Scholar 

  35. A. Vardy, Trellis structure of codes, in Handbook of Coding Theory, Volume II, (V. S. Pless and W. C. Huffman, eds.), North-Holland, Amsterdam, The Netherlands (1998).

    Google Scholar 

  36. V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inform. Theory, Vol. IT–37, No. 5 (1991) pp. 1412–1418.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, National Chi-Nan University, Nantou, Taiwan, 545

    Houshou Chen

  2. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan, 48109-2122

    John T. Coffey

Authors
  1. Houshou Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John T. Coffey
    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

Chen, H., Coffey, J.T. Trellis Structure and Higher Weights of Extremal Self-Dual Codes. Designs, Codes and Cryptography 24, 15–36 (2001). https://doi.org/10.1023/A:1011265011802

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011265011802

Search

Navigation