Advertisement

Trace-function on a Galois ring in coding theory

  • Alexander A. Nechaev
  • Alexey S. Kuzmin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1255)

Abstract

Patterns of the distribution of elements in words of linear codes over a Galois ring and in their representations over a Galois field are investigated. Often they may be evaluated using numbers of some special solutions of the equation defined by the trace-function on a Galois ring. Here the solutions of such an equation over a Galois ring R=GR(q2,4) of characteristic 4 are enumerated. It allows us in particular to describe the complete weight enumerators of the base linear code K R (m) and the appropriate Kerdock code K q (m+1) over a Galois field of q=21 elements.Results based on properties of special quadrics over GF(21) arise by description of the 2-adic decomposition of the trace-function.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nechaev A. A., Kuzmin A. S. Linearly presentable codes, Proceedings of the 1996 IEEE Int. Symp. Inf. Theory and Appl., Victoria B. C., Canada, 1996, pp. 31–34.Google Scholar
  2. 2.
    Nechaev A. A. Trace function in Galois ring and noise stable codes (in Russian), V All-Union Symp. on theory of rings, algebras and modules, Novosibirsk, p. 97, 1982.Google Scholar
  3. 3.
    Nechaev A. A. Kerdock code in a cyclic form (in Russian). Diskr. Math. (USSR), 1 (1989), No. 4, 123–139. English translation: Diskrete Math. and Appl., 1 (1991), No. 4, 365–384 (VSP).Google Scholar
  4. 4.
    Hammons A. R., Kumar P. V., Calderbank A. R., Sloane N. J. A., Solé P. The ℤ4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inf. Theory, vol. 40, No. 2, pp. 301–319, 1994.Google Scholar
  5. 5.
    Nechaev A. A., Kuzmin A. S. ℤ4-linearity, two approaches, V-th Int. Workshop on Alg. and Comb. Coding Theory, Proceedings, Sozopol, Bulgaria, pp. 212–215, 1996.Google Scholar
  6. 6.
    Kurakin V. L., Kuzmin A. S., Mikhalev A. V., Nechaev A. A. Linear recurrences over rings and modules. J. of Math. Science. Contemporary Math. and it's Appl. Thematic surveys, vol. 10, 1994, J. of Math. Sciences, vol. 76, No. 6, pp. 2793–2915, 1995.Google Scholar
  7. 7.
    Kuzmin A. S., Nechaev A. A. Linearly presented codes and Kerdock code over an arbitrary Galois field of the characteristic 2, Russian Math. Surveys, vol. 49, No. 5. 1994.Google Scholar
  8. 8.
    Nechaev A. A. Linear codes over modules and over spaces: MacWilliams identity, Proceedings of the 1996 IEEE Int. Symp. Inf. Theory and Appl., Victoria B. C., Canada, 1996, pp. 35–38.Google Scholar
  9. 9.
    Kuzmin A. S., Nechaev A. A. Linear recurring sequences over Galois rings, Algebra and Logic, Plenum Publ. Corp., 34 (1995), No. 2.Google Scholar
  10. 10.
    Lidl R., Niederreiter H. Finite fields, Addison-Wesley, London, 1983.Google Scholar
  11. 11.
    Diedonné J.-A. La Géométrie des Groupes Classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5, Springer, 1971.Google Scholar
  12. 12.
    Seroussi G., Lempel A. Factorization of symmetric matrices and trace-orthogonal bases in finite fields. SIAM J. Computing, vol. 9, pp. 758–767, 1980.Google Scholar
  13. 13.
    MacWilliams F. J., Sloane N. J. A. The Theory of Error-Correcting Codes, North-Holland Publ. Co., 1977.Google Scholar
  14. 14.
    Yang K., Helleseth T., Kumar P. V., Shanbhag A. G. On the weight hierarchy of Kerdock codes over ℤ4. IEEE Trans. on Inf. Th., vol. 42, No. 5, Sept. 1996, pp. 1587–1593.Google Scholar
  15. 15.
    Delsarte P., Göthals J.-M. Alternating bilinear forms over GF(q). J. Combin. Theory, 19 A, 1975, pp. 26–50.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Alexander A. Nechaev
    • 1
  • Alexey S. Kuzmin
    • 1
  1. 1.Center of New Informational Technologies of Moscow State UniversityRussia

Personalised recommendations