Advertisement

Eurocode ’92 pp 159-171 | Cite as

Bounds for Self-Complementary Codes and Their Applications

  • V. I. Levenshtein
Part of the International Centre for Mechanical Sciences book series (CISM, volume 339)

Abstract

Self-complementary codes in the Hamming space, i.e., binary codes that together with any vector contain its complement, are considered. The bound on the size of a self-complementary code with a given minimum distance d presented here is in general better than the corresponding bound for arbitrary binary codes in the Hamming space. Some applications of this bound for estimating the minimum distance of self-dual binary codes, the cross-correlation of arbitrary binary codes, the modulus of sums of Legendre symbols of polynomials, and some parameters of randomness properties of binary codes, are given.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AGHP]
    Alon N., Goldreich O., Hastad J., Peralta R., Simple constructions of almost k-wise independent random variables, Proc. of the 31st Annual Symposium on the Foundations of Computer Science, 1991.Google Scholar
  2. [BI]
    Bannai E., Ito T., Algebraic Combinatorics I, Association Schemes, Benjamin/Cummings, London, 1984.Google Scholar
  3. [BCN]
    Brouwer A.E., Cohen A.M., Neumaier A., Distance-regular graphs, Springer-Verlag, Berlin, 1989.CrossRefzbMATHGoogle Scholar
  4. [CS]
    Conway J.H., Sloane N.J.A., A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, IT-36 (1990), 1319–1333.Google Scholar
  5. [D]
    Delsarte Ph., An algebraic approach to the association schemes of coding theory,Philips Res. Reports Suppl. 10 (1973).Google Scholar
  6. [L1]
    Levenshtein V.I., On choosing polynomials to:4btain bounds in packing problems (in Russian), in Proc. Seventh All-Union Conf. on Coding Theory and Information Transm., Part II, Moscow-Vilnius, 1978, 103–108.Google Scholar
  7. [L2]
    Levenshtein V.I., Bounds to the maximum size of code with limited scalar product modulus,Soviet Math. Doklady, vol. 25 (1982), N.2, 525–531.Google Scholar
  8. [L3]
    Levenshtein V.I., Bounds for packings of metric spaces and some their applications (in Russian),Problemy Kiberneticki, Issue 40, Moscow, “Nauka” 1983, 43–110.Google Scholar
  9. [L4]
    Levenshtein V.I., Designs as maximal codes in polynomial metric spaces,Acta Applicandae Mathematicae, 29, (1992), 1–82.Google Scholar
  10. [MRRW]
    McEliece R.J., Rodemich E.R., Rumsey H., jr., and Welch L.R., New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities, IEEE Trans. Inform. Theory, IT-23 (1977), 157–166.Google Scholar
  11. [MS]
    MacWilliams F.J., Sloane N.J.A., The theory of error-correcting codes, North Holland Publ. Co., Amsterdam, 1977.Google Scholar
  12. [S1]
    Sidelnikov V.M., On mutual correlation of sequences (in Russian),Problemy Kiberneticki, Issue 24, Moscow, “Nauka” 1971, 15–42 (a short description in English in Soviet Math. Doklady, 12, N1 (1971), 197–201).Google Scholar
  13. [S2]
    Sidelnikov V.M., On extremal polynomials used to estimate the size of codes,Problems of Information Transmission, 16, N3 (1980), 174–186.Google Scholar
  14. [SZ]
    Szego G. Orthogonal polynomials, Vol. XXII, AMS Col. Pub., Providence, Rhode Island, 1939.Google Scholar

Copyright information

© Springer-Verlag Wien 1993

Authors and Affiliations

  • V. I. Levenshtein
    • 1
  1. 1.Russian Academy of SciencesMoscowRussia

Personalised recommendations