Advertisement

The BM Algorithm and the BMS Algorithm

  • Shojiro Sakata
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 485)

Abstract

We present a brief overview of the Berlekamp-Massey (BM) algorithm and of the Berlekamp-Massey-Sakata (BMS) algorithm, with an emphasis on a close relationship between them, which can be summarized as follows: BM algorithm + Gröbner basis = BMS algorithm. Furthermore, we describe various forms of these algorithms, all of which are related to fast decoding of algebraic codes including algebraic-geometric (AG) codes. As a byproduct, we obtain a recursive BMS algorithm which is an extension of Blahut’s recursive BM algorithm. Throughout this paper, we recall the assertions of R.E. Blahut on the importance of fast decoding for good codes, particularly as described in his books, which have spurred recent developments of novel decoding algorithms of AG codes.

Keywords

Total Order Polynomial Vector Linear Feedback Shift Register Linear Recurrence Algebraic Code 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.E. Blahut, Theory and Practice of Error Control Codes, Chapter 11: Fast Algorithms, pp. 308–346, Reading, MA: Addison Wesley 1983.MATHGoogle Scholar
  2. [2]
    —, Algebraic Methods for Signal Processing and Communications Coding, New York: Springer-Verlag 1992.CrossRefGoogle Scholar
  3. [3]
    E.R. Berlekamp, Algebraic Coding Theory, New York: McGraw-Hill 1968.MATHGoogle Scholar
  4. [4]
    J.L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, vol. 15, pp. 122–127, 1969.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    S. Sakata, Finding a minimal set of linear recurring relations capable of generating a given finite 2D cyclic codes and their duals, J. Symbolic Computation, vol. 5, pp. 1321–1337, 1988.MathSciNetCrossRefGoogle Scholar
  6. [6]
    —, Extension of the BM algorithm to N dimensions, Information and Computation, vol. 84, pp. 207–239, 1990.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    —, A vector version of the BMS algorithm for implementing fast erasure-and-error decoding of one-point AG codes, submitted to Appl. Algebra Engrg. Comm. Comput. Google Scholar
  8. [8]
    S. Sakata, H. Elbrønd Jensen, and T. Høholdt, Generalized Berlekamp-Massey decoding of algebraic-geometric codes up to half the Feng-Rao bound,” IEEE Trans. Inform. Theory, vol.41, pp. 1762–1769, 1995.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    S. Sakata, J. Justesen, Y. Madelung, H. Elbrønd Jensen, and T. Høholdt, Fast decoding of algebraic geometric codes up to the designed distance, IEEE Trans. Inform. Theory, vol.41, pp. 1672–1677, 1995.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    S. Sakata and M. Kurihara, A systolic array architecture for implementing a fast parallel decoding algorithm of one-point AG codes, submitted to IEEE Trans. Inform. Theory. Google Scholar
  11. [11]
    S. Sakata, D. Leonard, H. Elbrøn d Jensen, and T. Høholdt, Fast erasure-and-error decoding of AG codes up to the Feng-Rao bound, submitted to IEEE Trans. Inform. Theory. Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Shojiro Sakata
    • 1
  1. 1.Department of Computer Science and MathematicsUniversity of Electro-CommunicationsChofu-ShiJapan

Personalised recommendations