Parameters of Goppa codes revisited

IEEE Transactions on Information Theory

Volumn 43, Issue 6, 1997, Pages 1811-1819

  Levy dit Vehel, Françoise ^b,c   Litsyn, Simon ^a,d  

^a IEEE

^b INRIA   (France)

^c GEMPLUS   (France)

^d TEL AVIV UNIVERSITY   (Israel)

Author keywords

Covering radius; Distance distribution; Exponential sums; Generalized weights; Goppa codes

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; PARAMETER ESTIMATION; POLYNOMIALS;

COVERING RADIUS; DISTANCE DISTRIBUTION; EXPONENTIAL SUMS METHOD; GENERALIZED HAMMING WEIGHTS; GOPPA CODES;

BINARY CODES;

EID: 0031269351     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.641547     Document Type: Article

References (40)

1. E. R. Berlekamp, "Goppa codes," IEEE Trans. Inform. Theory, vol. IT-19, pp. 590-592, 1973.
2. E. R. Berlekamp and O. Moreno, "Extended double-error correcting binary Goppa codes are cyclic," IEEE Trans. Inform. Theory, vol. IT-19, pp. 817-818, 1973.
3. S. V. Bezzateev and N. A. Shekhunova, "On the constructive distance of the best known (55, 16, 16) Goppa code," Probl. Pered. Inform., vol. 23, no. 4, p. 119, 1987.
4. _, "Subclass of binary Goppa codes with minimal distance equal to the designed distance," IEEE Trans. Inform. Theory, vol. 41, pp. 554-555, Mar. 1995.
5. S. V. Bezzateev, N. A. Shekhunova, and E. T. Mironchikov, "A subclass of binary Goppa codes," Probl. Inform. Transm., vol. 25, no. 3, pp. 98-102, 1989.
6. C. L. Chen, "Equivalent irreducible Goppa codes," IEEE Trans. Inform. Theory, vol. IT-24, pp. 766-769, Nov. 1978.
7. G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering Codes. Amsterdam, The Netherlands: Elsevier, 1997.
8. G. Cohen, S. Litsyn, A. Lobstein, and H. F. Mattson, Jr., "Covering radius 1985-1994," Applicable Alg. in Eng., Commun. and Comput., vol. 8, no. 67, 1997.
9. G. Cohen, S. Litsyn, and G. Zémor, "Upper bounds on generalized distances," IEEE Trans. Inform. Theory, vol. 40, pp. 2090-2092, Nov. 1994.
10. G. D. Forney, "Dimension/length profiles and trellis complexity of linear block codes," IEEE Trans. Inform. Theory, vol. 40, pp. 1741-1752, Nov. 1994.
11. V. D. Goppa, "A new class of linear error-correcting codes," Probl. Inform. Transm., vol. 6, pp. 207-212, 1970.
12. _, "Rational representation of codes and (L, g)-codes," Probl. Inform. Transm., vol. 7, pp. 223-229, 1971.
13. _, "Codes constructed on the basis of (L, g)-codes," Probt. Inform. Transm., vol. 8, pp. 165-166, 1972.
14. T. Helleseth, "On the covering radius of cyclic linear codes and arithmetic codes," Discr. Appl. Math., vol. 11, pp. 157-173, 1985.
15. T. Helleseth, T. Kløve, and Ø. Ytrehus, "Generalized Hamming weights of linear codes," IEEE Trans. Inform. Theory, vol. 38, pp. 1133-1140, 1992.
16. G. Kabatiansky, "On the second generalized Hamming weight," in Int. Workshop on Algebraic and Combinatorial Coding Theory (Bulgaria, 1992), pp. 98-100.
17. I. Krasikov and S. Litsyn, "On spectra of BCH codes," IEEE Trans. Inform. Theory, vol. 41, pp. 786-788, May 1995.
18. G. Lachaud and J. Wolfmann, "The weights of the orthogonals of the extended quadratic binary Goppa codes," IEEE Trans. Inform. Theory, vol. 36, pp. 686-692, 1990.
19. F. Levy-dit-Vehel and S. Litsyn, "On the covering radius of long Goppa codes," in Proc. AAECC'11, Lecture Notes in Computer Science, no. 948, Berlin, Germany: Springer-Verlag, 1995, pp. 341-346.
20. M. Loeloeian and J. Conan, "A transform approach to Goppa codes," IEEE Trans. Inform. Theory, vol. IT-33, pp. 105-115, 1987.
21. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
22. D. M. Mandelbaum, "On the derivation of Goppa codes," IEEE Trans. Inform. Theory, vol. IT-21, pp. 110-111, 1975.
23. C. J. Moreno and O. Moreno, "Exponential sums and Goppa codes I," Proc. Amer. Math. Soc., vol. 111, pp. 523-531, 1991.
24. _, "Exponential sums and Goppa codes II," IEEE Trans. Inform. Theory, vol. 38, pp. 1222-1229, July 1992.
25. C. T. Retter, "Bounds on Goppa codes," IEEE Trans. Inform. Theory, vol. IT-22, pp. 476-482, July 1976.
26. _, "Intersecting Goppa codes," IEEE Trans. Inform. Theory, vol. 35, pp. 822-828, July 1989.
27. T. J. Rivlin, Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. New York: Wiley, 1992.
28. A. M. Roseiro, J. I. Hall, J. E. Adney, and M. Siegel, "The trace operator and redundancy of Goppa codes," IEEE Trans. Inform. Theory, vol. 38, pp. 1130-1133, 1992.
29. Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa, "Further results on Goppa codes and their applications to constructing efficient binary codes," IEEE Trans. Inform. Theory, vol. IT-22, pp. 518-526, Sept. 1976.
30. A. Tietäväinen, "On the covering radius of long binary BCH codes," Discr. Appl. Math., vol. 16, pp. 75-77, 1987.
31. M. A. Tsfasman and S. G. Vlǎduţ, "Geometric approach to higher weights," IEEE Trans. Inform. Theory, vol. 41, pp. 1564-1588, Nov. 1995.
32. K. K. Tzeng and C. Y. Yu, "Characterization theorems for extending Goppa codes to cyclic codes," IEEE Trans. Inform. Theory, vol. IT-25, pp. 246-250, Mar. 1979.
33. K. K. Tzeng and K. P. Zimmerman, "On extending Goppa codes to cyclic codes," IEEE Trans. Inform. Theory, vol. IT-21, pp. 712-716, 1975.
34. G. Van der Geer and M. Van der Vlugt, "On generalized Hamming weights of BCH codes," IEEE Trans. Inform. Theory, vol. 40, pp. 543-546, Mar. 1994.
35. M. Van der Vlugt, "The true dimension of certain binary Goppa codes," IEEE Trans. Inform. Theory, vol. 36, pp. 397-398, Mar. 1990.
36. J. H. Van Lint, Introduction to Coding Theory. New York: Springer-Verlag, 1992.
37. A. L. Vishnevetsky, "On the cyclicity of the extended Goppa codes," Probl. Pered. Inform., vol. 18, no. 3, pp.14-18, 1982.
38. S. G. Vlǎduţ and A. N. Skorobogatov, "Weight distributions of subfield subcodes of algebraic-geometric codes," Probl. Inform. Transm., vol. 1, pp. 19-29, 1991.
39. V. K. Wei, "Generalized Hamming weights for linear codes," IEEE Trans. Inform. Theory, vol. 37, pp. 1412-1418, 1991.
40. M. Wirtz, "On the parameters of Goppa codes," IEEE Trans. Inform. Theory, vol. 34, pp. 1341-1343, Sept. 1988.