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Designs, Codes and Cryptography

, Volume 63, Issue 3, pp 321–330 | Cite as

Weight enumeration of codes from finite spaces

  • Relinde P. M. J. Jurrius
Open Access
Article

Abstract

We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a finite projective or affine space. As a result from the geometric method we use for the weight enumeration, we also completely determine the set of supports of subcodes and words in an extension code.

Keywords

Coding theory Weight enumeration Finite geometry 

Mathematics Subject Classification (2000)

05B25 11T71 

Notes

Acknowledgments

The author would like to thank Vladimir Tonchev for coming up with the question about the weight enumerator of the extension codes of the Simplex code, and for his encouraging conversations on the subject.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Assmus E.F. Jr.: On the Reed-Muller codes. Discrete Math. 106/107, 25–33 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Assmus E.F. Jr., Key J.D.: Designs and their codes. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  3. 3.
    Calderbank R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Heijnen P., Pellikaan G.R.: Generalized Hamming weights of q-ary Reed-Muller codes. IEEE Trans. Inform. Theory 44, 181–196 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths n 1((q l−1)/n). Discrete Math. 18, 179–211 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Jurrius R.P.M.J., Pellikaan G.R.: Algebraic geometric modeling in information theory. In: Codes, arrangements and matroids. Series on Coding Theory and Cryptology. World Scientific Publishing, Hackensack, NJ (2011) http://www.worldscibooks.com/series/sctc_series.shtml.
  7. 7.
    Katsman G.L., Tsfasman M.A.: Spectra of algebraic-geometric codes. Probl. Inform. Transm. 23, 19–34 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kløve T.: The weight distribution of linear codes over GF(q l) having generator matrix over GF(q). Discrete Math. 23, 159–168 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kløve T.: Support weight distribution of linear codes. Discrete Math. 106/107, 311–316 (1992)CrossRefGoogle Scholar
  10. 10.
    Mphako E.G.: Tutte polynomials of perfect matroid designs. Combin. Probab. Comput. 9, 363–367 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Simonis J.: The effective length of subcodes. AAECC 5, 371–377 (1993)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tonchev V.D.: Linear perfect codes and a characterization of the classical designs. Des. Codes Cryptogr. 17, 121–128 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Tsfasman M.A., Vlǎdut S.G.: Algebraic-geometric codes. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  14. 14.
    Tsfasman M.A., Vlǎdut S.G.: Geometric approach to higher weights. IEEE Trans. Inform. Theory 41, 1564–1588 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Wei V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory 37, 1412–1418 (1991)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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