The weight distribution of double-error-correcting goppa codes

  • Arne Dür
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 307)


In this paper the weight distribution of binary Goppa codes with location set GF (2m) and irreducible quadratic Goppa polynomial is studied. These codes are all equivalent and have parameters [2m, 2m–2m, 5]. An explicit formula for the number of codewords of weight 5 and 6 is derived. The weight distribution of the dual codes is related to the weight distribution of a reversible irreducible cyclic code of length 2m+1 and dimension 2m whose weights can be expressed in terms of Kloosterman sums over GF (2m). As numerical examples the weight distribution of the double-error-correcting Goppa codes of block length 8,16,32, and 64 are computed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BAUMERT, L.D., and MCELIECE, R.J. (1972), Weights of irreducible cyclic codes, Inform. and Control 20, pp. 158–175.Google Scholar
  2. BAUMERT, L.D., and MYKKELTVEIT, J. (1973), Weight distributions of some irreducible cyclic codes, DSN Progress Report 16, pp. 128–131 (published by Jet Propulsion Laboratory, Pasadena, California).Google Scholar
  3. BERLEKAMP, E.R., and MORENO, O. (1973), Extended double-error-correcting binary Goppa codes are cyclic, IEEE Trans. Info. Theory 19, pp. 817–818.Google Scholar
  4. CARLITZ, L. (1969), Gauss sums over finite fields of order 2n, Acta Arithmetica 15, pp. 247–265.Google Scholar
  5. CARLITZ, L. (1969), Kloosterman sums and finite field extensions, Acta Arithmetica 16, pp. 179–193.Google Scholar
  6. CARLITZ, L., and UCHIYAMA, S. (1957), Bounds for exponential sums, Duke Math. J. 24, pp. 37–41.Google Scholar
  7. COHEN, G.D., and GODLEWSKI, P.J. (1975), Residual error rate of binary linear block codes, in: CISM Lecture Notes 219 (G. Longo, ed.), pp. 325–335, Springer, Wien-New York.Google Scholar
  8. FENG, G.L., and TZENG, K.K. (1984), On quasi-perfect property of double-error-correcting Goppa codes and their complete decoding, Inform. and Control 61, pp. 132–146.Google Scholar
  9. LIDL, R., and NIEDERREITER, H. (1983), Finite Fields, Addison-Wesley, Reading, Massachusetts.Google Scholar
  10. MACWILLIAMS, F.J., and SEERY, J. (1981), The weight distributions of some minimal cyclic codes, IEEE Trans. Info. Theory 27, pp. 796–806.Google Scholar
  11. MACWILLIAMS, F.J., and SLOANE, N.J.A. (1977), The Theory of Error-Correcting Codes, North Holland, Amsterdam.Google Scholar
  12. McELIECE, R.J. (1975), Irreducible cyclic codes and Gauss sums, in: Combinatorics (M. Hall, Jr., and J.H. van Lint, eds.), pp. 185–202, Reidel, Dordrecht-Boston.Google Scholar
  13. McELIECE, R.J. (1980), Correlation properties of sets of sequences derived from irreducible cyclic codes, Inform. and Control 45, pp. 18–25.Google Scholar
  14. McELIECE, R.J., and Rumsey, H. (1972), Euler products, cyclotomy, and coding, J. Number Theory 4, pp. 302–311.Google Scholar
  15. MORENO, O. (1981), Goppa codes related quasi-perfect double-error-correcting codes, in "Abstracts of Papers", IEEE Internat. Sympos. Inform. Theory, Santa Monica, California.Google Scholar
  16. NIEDERREITER, H. (1977), Weights of cyclic codes, Inform. and Control 34, pp. 130–140.Google Scholar
  17. TZENG, K.K., and ZIMMERMANN, K. (1975), On extending Goppa codes to cyclic codes, IEEE Trans. Info. Theory 21, pp. 712–716.Google Scholar
  18. VAN LINT, J.H. (1982), Introduction to Coding Theory, Springer, New York-Heidelberg-Berlin.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Arne Dür
    • 1
  1. 1.Institut für MathematikUniversität InnsbruckInnsbruckAustria

Personalised recommendations