Designs, Codes and Cryptography

, Volume 17, Issue 1–3, pp 269–288 | Cite as

On Z4-Linear Goethals Codes and Kloosterman Sums

  • Tor Helleseth
  • Victor Zinoviev
Article

Abstract

Studying the coset weight distributions of the Z4-linear Goethals codes, e connect these codes with the Kloosterman sums. From one side, e obtain for some cases, of the cosets of weight four, the exact expressions for the number of code ords of weight four in terms of the Kloosterman sums. From the other side, e obtain some limitations for the possible values of the Kloosterman sums, hich improve the well known results due to Lachaud and Wolfmann kn:lac.

Z4-linear Goethals code coset weight distribution nonlinear system of equations exponential sums Kloosterman sums 

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References

  1. 1.
    L. Carlitz, Kloosterman sums and finite field extensions, Acta Arithmetica, Vol. XV,No. 2 (1969) pp. 179-193.Google Scholar
  2. 2.
    L. Carlitz and S. Uchiyama, Bounds for exponential sums, Duke Math. Journal, Vol. 24 (1957) pp. 37-42.Google Scholar
  3. 3.
    P. Charpin, Personal Communication, 1998.Google Scholar
  4. 4.
    P. Charpin and V. A. Zinoviev, On coset weight distributions of the 3-error-correcting BCH codes, SIAM J. of Discrete Math., Vol. 10,No. 1 (February 1997) pp. 128-145.Google Scholar
  5. 5.
    J. F. Dillon, Elementary Hadamard difference sets, Proc. Sixth SECCGTC (F. Hoffman et al., eds.), Utilitas Math., Winnipeg (1975).Google Scholar
  6. 6.
    T. A Dowling and R. McEliece, Cross-correlations of Reverse Maximal-Length Shift Register Sequences, Jet Propulsion Lab., Pasadena, California, Space Programs Summary, 37–53, Vol. III, pp. 192-193.Google Scholar
  7. 7.
    J. M. Goethals, Two dual families of nonlinear binary codes, Electron. Lett., Vol. 10,No. 23 (1974) pp. 471-472.Google Scholar
  8. 8.
    R. Hammons, P. V. Kumar, N. J. A. Sloane, R. Calderbank and P. Solé, The Z 4-Linearity of Kerdock, Preparata, Goethals, and Related Codes, IEEE Trans. on Inform. Theory, Vol. 40 (1994) pp. 301-319.Google Scholar
  9. 9.
    T. Helleseth and P. V. Kumar, The Algebraic Decoding of the Z 4-Linear Goethals Code, IEEE Trans. on Inform. Theory, Vol. 41,No. 6 (November 1995) pp. 2040-2048.Google Scholar
  10. 10.
    T. Helleseth and V. Zinoviev, On coset weight distributions of the Z 4-linear Goethals Codes, IEEE Trans. on Inform. Theory (submitted).Google Scholar
  11. 11.
    G. Lachaud and J. Wolfmann, The Weights of the Orthogonals of the Extended Quadratic Binary Goppa Codes, IEEE Trans. on Inform. Theory, Vol. 36,No. 3 (May 1990) pp. 686-692.Google Scholar
  12. 12.
    R. Lidl and H. Niederreiter, “Finite Fields,” Encyclopedia of Mathematics and Its Applications, Vol. 20, Addison Wesley, Reading, MA (1983).Google Scholar
  13. 13.
    F. J. Macwilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland (1986).Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Tor Helleseth
    • 1
  • Victor Zinoviev
    • 2
  1. 1.Department of InformaticsUniversity of BergenBergenNoray
  2. 2.Institute for Problems of Information Transmission of theRussian Academy of SciencesMoscoRussia

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