On Fast Interpolation Method for Guruswami-Sudan List Decoding of One-Point Algebraic-Geometry Codes

  • Shojiro Sakata
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2227)


Fast interpolation methods for the original and improved versions of list decoding of one-point algebraic-geometry codes are presented. The methods are based on the Gröbner basis theory and the BMS algorithm for multiple arrays, although their forms are different in the original list decoding algorithm (Sudan algorithm) and the improved list decoding algorithm (Guruswami-Sudan algorithm). The computational complexity is less than that of the conventional Gaussian elimination method.


Total Order Pole Order Multiple Array List Decode Receive Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Sudan, “Decoding of Reed-Solomon codes beyond the error-correction bound,” J. Complexity, 13, 180–193, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. A. Shokrollahi, H. Wasserman, “List decoding of Algebraic-geometric codes,” IEEE Trans. Inform. Theory, 45, 432–437, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    V. Guruswami, M. Sudan, “Improved decoding of Reed-Solomon and algebraicgeometry codes,” IEEE Trans. Inform. Theory, 45, 1757–1767, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. M. Roth, G. Ruckenstein, “Efficient decoding of Reed-Solomon codes beyond half the minimum distance,” IEEE Trans. Inform. Theory, 46, 246–257, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Y. Numakami, M. Fujisawa, S. Sakata, “A fast interpolation method for list decoding of Reed-Solomon codes,” Trans. IEICE, J83-A, 1309–1317, 2000. (in Japanese)Google Scholar
  6. 6.
    V. Olshevsky, M. A. Shokrollahi, “A displacement approach to efficient decoding of algebraic-geometric codes,” Proc. 31st ACM Symp. Theory of Comput., 235–244, 1999.Google Scholar
  7. 7.
    D. Augot, L. Pecquet, “A Hensel lifting to replace factorization in list-decoding of algebraic-geometric and Reed-Solomon codes,” IEEE Trans. Inform. Theory, 46, 2605–2614, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Matsumoto, “On the second step in the Guruswami-Sudan list decoding for AG codes,” Tech. Report of IEICE, IT99-75, 65–70, 2000.Google Scholar
  9. 9.
    X. W. Wu, P. H. Siegel, “Efficient list decoding of algebraic-geometric codes beyond the error correction bound,” preprint.Google Scholar
  10. 10.
    Sh. Gao, M. A. Shokrollahi, “Computing roots of the polynomials over function fields of curves,” in D. Joyner (Ed.), Coding Theory & Crypt., Springer, 214–228, 2000.Google Scholar
  11. 11.
    S. Sakata, “N-dimensional Berlekamp-Massey algorithm for multiple arrays and construction of multivariate polynomials with preassigned zeros,” in T. Mora (Ed.), Appl. Algebra, AlgebraicA lgorithms & Error-Correcting Codes, LNCS: 357, Springer, 356–376, 1989.Google Scholar
  12. 12.
    S. Sakata, Y. Numakami, “A fast interpolation method for list decoding of RS and algebraic-geometric codes,” presented at the 2000 IEEE Intern. Symp. Inform. Theory, Sorrento, Italy, June 2000.Google Scholar
  13. 13.
    S. Sakata, “Extension of the Berlekamp-Massey algorithm to N dimensions,” Inform. & Comput., 84, 207–239, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Nielsen, T. Høholdt, “Decoding Reed-Solomon codes beyond half the minimum distance,” in J. Buchmenn, T. Høholdt, T. Stichtenoth, H. Tapia-Recillas (Eds.), Coding Theory, Crypt. & Related Areas, Springer, 221–236, 2000.Google Scholar
  15. 15.
    T. Høholdt, R. Nielsen, “Decoding Hermitian codes with Sudan’s algorithm,” preprint.Google Scholar
  16. 16.
    W. Feng, R. E. Blahut, “Some results of the Sudan algorithm” presented at the 1998 IEEE Intern. Symp. Inform. Theory, Cambridge, USA, August 1998.Google Scholar
  17. 17.
    S. Sakata, “On determining the independent point set for doubly periodic arrays and encoding two-dimensional cyclic codes and their duals,” IEEE Trans. Inform. Theory, 27, 556–565, 1981.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Shojiro Sakata
    • 1
  1. 1.Department of Information and Communication EngineeringThe University of Electro-CommunicationsTokyoJapan

Personalised recommendations