Advertisement

An improved upper bound on covering radius

  • H. F. MattsonJr.
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 228)

Abstract

A simple upper bound on covering radius yields new information on various codes. It leads us to show that the nonlinear codes of Sloane and Whitehead [18] are quasi-perfect. We get some new bounds for the Berlekamp-Gale switching problem [7]. It gives the exact covering radius for some codes of length up to 31 and is within 1 or 2 of the exact value for the even quadratic-residue codes of lengths 41 and 47.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. F. Assmus, Jr., and Vera Pless, “On the covering radius of extremal self-dual codes,” IEEE Trans. Inform. Theory, IT-29 (1983) 359–363.Google Scholar
  2. 2.
    E. R. Berlekamp Algebraic Coding Theory, McGraw Hill, New York, 1968.Google Scholar
  3. 3.
    Gérard D. Cohen, Mark R. Karpovsky, H. F. Mattson, Jr., and James R. Schatz, “Covering Radius—Survey and Recent Results,” IEEE Trans. Inform. Theory IT-31 (1985), 328–343.Google Scholar
  4. 4.
    Ph. Delsarte, “Four fundamental parameters of a code and their combinatorial significance,” Inform. and Control 23 (1973), 407–438.Google Scholar
  5. 5.
    Diane E. Downey and N. J. A. Sloane, “The covering radius of cyclic codes of length up to 31,” IEEE Trans. Inform. Theory, IT-31 (1985), 446–447.Google Scholar
  6. 6.
    M. J. E. Golay, “Binary coding,” IEEE Trans. Inform. Theory, PGIT-4 (1954) 23–28.Google Scholar
  7. 7.
    R. L. Graham and N. J. A. Sloane, “On the covering radius of codes,” IEEE Trans. Inform. Theory, IT-31 (1985), 385–401.Google Scholar
  8. 8.
    J. H. Griesmer, “A bound for error-correcting codes,” IBM J. Res. Develop. 4 (1960), 532–542.Google Scholar
  9. 9.
    H. J. Helgert and R. D. Stinaff, “Minimum-distance bounds for binary linear codes,” IEEE Trans. Inform. Theory IT-19 (1973), 344–356.Google Scholar
  10. 10.
    D. Julin, “Two improved block codes,” IEEE Trans. Inform. Theory, IT-11 (1965) 459.Google Scholar
  11. 11.
    Mark R. Karpovsky, public communication, at Journée sur le rayon de recouvrement et codes correcteurs d'érreurs, ENST, Paris, 26 June 1984.Google Scholar
  12. 12.
    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977.Google Scholar
  13. 13.
    H. F. Mattson, Jr., “An upper bound on covering radius,” Annals of Discrete Math. 17 (1982) 453–458.Google Scholar
  14. 14.
    H. F. Mattson, Jr. “Another upper bound on covering radius,” IEEE Trans. Inform. Theory, IT-29 (1983) 356–359.CrossRefGoogle Scholar
  15. 15.
    W. W. Peterson and E. J. Weldon, Jr., Error-correcting codes, Second Edition, Cambridge, M.I.T. 1972.Google Scholar
  16. 16.
    V. Pless and E. A. Prange, “Weight distribution of all cyclic codes ... [of length] 31 over GF(2)” unpublished memorandum, September, 1962.Google Scholar
  17. 17.
    James R. Schatz, “On the coset leaders of Reed-Muller codes,” Ph. D. dissertation, Syracuse University, 1979.Google Scholar
  18. 18.
    Neil J. A. Sloane and Donald S. Whitehead, “New family of single-error correcting codes, IEEE Trans. Inform. Theory, IT-16 (1970) 717–719.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • H. F. MattsonJr.
    • 1
  1. 1.School of Computer and Information ScienceSyracuse UniversitySyracuse

Personalised recommendations