Skip to main content
SpringerLink
Log in

A Generalized Rational Interpolation Problem and the Solution of the Welch–Berlekamp Key Equation

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We show that an algorithm designed to solve the Welch–Berlekamp key equation may also be used to solve a more general problem, which can be regarded as a finite analogue of a generalized rational interpolation problem. As a consequence, we show that a single algorithm exists which can solve both Berlekamp's classical key equation (usually solved by the Berlekamp–Massey algorithm) and the Welch–Berlekamp key equation which arise in the decoding of Reed–Solomon codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. C. Antoulas, Recursive modelling of discrete time series, in Linear Algebra for Control Theory, Inst. Math. and Applic., Springer, Berlin, 62 (1994) pp. 1–20.

    Google Scholar 

  2. A. C. Antoulas and B. D. Q. Anderson, On the scalar rational interpolation problem, IMA J. Math. Control and Inform., Vol. 3 (1986) pp. 61–88.

    Google Scholar 

  3. A. C. Antoulas, J. A. Ball, J. Kang and J. C. Willems, On the solution of the minimal rational interpolation problem, Linear Algebra Appl., Vol. 137/138 (1990) pp. 511–573.

    Google Scholar 

  4. A. C. Antoulas and J. C. Willems, A behavioural approach to exact linear modelling, IEEE Trans. Automatic Control, Vol. 38,No. 2 (1993), pp. 1776–1802.

    Google Scholar 

  5. E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York (1968).

    Google Scholar 

  6. E. R. Berlekamp, Bounded distance + 1 soft decision Reed-Solomon decoding, (preprint).

  7. G. D. Carter, Aspects of linear complexity, PhD thesis, University of London (1989).

  8. W. G. Chambers, R. E. Peile, K. Y. Tsie and N. Zein, Algorithm for solving the Welch-Berlekamp key equation, with a simplified proof, Electronics Letters, Vol. 29,No. 18 (1993) pp. 1620–1621.

    Google Scholar 

  9. D. Dabiri and I. F. Blake, Fast parallel algorithms for decoding Reed-Solomon codes based on remainder polynomials, IEEE Trans. Inform. Theory, Vol. IT-41,No. 4 (1995) pp. 873–885.

    Google Scholar 

  10. P. Fitzpatrick, On the key equation, IEEE Trans. Inform. Theory, Vol. IT-41,No. 5 (1995) pp. 1–13.

    Google Scholar 

  11. T. H. Liu, A new decoding algorithm for Reed-Solomon codes, PhD thesis, University of Southern California, Los Angeles, CA (1984).

    Google Scholar 

  12. J. L. Massey, Shift register synthesis and BCH decoding, IEEE Trans. Inform. Theory, Vol. IT-15 (1969) pp. 122–127.

    Google Scholar 

  13. M. Morii and M. Kasahara, Generalized key-equation of remainder decoding algorithm for Reed-Solomon codes, IEEE Trans. Inform. Theory, Vol. IT-38 (1992) pp. 1801–1807.

    Google Scholar 

  14. R. E. Peile, On the performance and complexity of a generalized minimum distance Reed-Solomon decoding algorithm, International Journal of Satellite Communications, Vol. 12,No. 4 (1994) pp. 333–359.

    Google Scholar 

  15. L. Welch and E. R. Berlekamp, Error correction for algebraic block codes, U.S. Patent 4 633 470 (1983).

Download references

Author information

Authors and Affiliations

  1. Information Security Group, Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom

    Simon R. Blackburn

Authors
  1. Simon R. Blackburn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blackburn, S.R. A Generalized Rational Interpolation Problem and the Solution of the Welch–Berlekamp Key Equation. Designs, Codes and Cryptography 11, 223–234 (1997). https://doi.org/10.1023/A:1008290027201

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008290027201

Search

Navigation