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Designs, Codes and Cryptography

, Volume 10, Issue 3, pp 341–350 | Cite as

Codes of Small Defect

  • A. Faldum
  • W. Willems
Article

Abstract

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.

linear codes defect of codes almost MDS codes Steiner systems weight hierarchy 

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References

  1. 1.
    E. F. Assmus, Jr. and H. F. Mattson, Jr., On weights in quadratic-residue codes, Discrete Mathematics, Vol. 3 (1972) pp. 1–20.Google Scholar
  2. 2.
    M. A. de Boer, Almost MDS codes, to appear in Designs, Codes and Cryptography.Google Scholar
  3. 3.
    S. M. Dodunekov and I. N. Landgev, On near-MDS codes, report LiTH-ISY-R-1563, Dept. of Electrical Engineering, Linköping University (1994).Google Scholar
  4. 4.
    J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York (1982).Google Scholar
  5. 5.
    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, sixth printing (1988).Google Scholar
  6. 6.
    S. Roman, Coding and Information Theory, Springer-Verlag, New York (1992).Google Scholar
  7. 7.
    M. A. Tsfasman and S. G. Vlăduţ, Algebraic-Geometric Codes, Kluwer Academic Publishers, Dordrecht (1991).Google Scholar
  8. 8.
    V. K. Wei, Generalized hamming weights for linear codes, IEEE Trans. Inf. Theory, Vol. 37 (1991) pp. 1412–1418.Google Scholar
  9. 9.
    A. Faldum and W. Willems, A characterization of codes with extreme parameters, to appear IEEE Trans. Inf. Theory, 42 (1996).Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • A. Faldum
    • 1
  • W. Willems
    • 1
  1. 1.Fakultät für MathematikUniversität MagdeburgMagdeburg

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