Mathematical Notes

, Volume 79, Issue 1–2, pp 41–54 | Cite as

Berlekamp—Massey Algorithm, Continued Fractions, Pade Approximations, and Orthogonal Polynomials

  • S. B. Gashkov
  • I. B. Gashkov


The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.

Key words

Berlekamp—Massey algorithm Pade approximations continued fraction orthogonal polynomial Laurent series Euclid's algorithm 


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  1. 1.
    D. Jungnickel, Finite Fields: Structure and Arithmetic, Wissenschaftsverlag, Mannheim-Leipzig-Wien, 1993.Google Scholar
  2. 2.
    R. Blahut, Theory and Practice of Error Control Codes, Addison-Wesley, Reading, MA, 1983; Russian translation: Mir, Moscow, 1986.Google Scholar
  3. 3.
    J. L. Massey, “Feedback Shift Register Synthesis and BCH decoding,” IEEE Trans. Inform. Theory, IT-15 (1969), 122–128.MathSciNetGoogle Scholar
  4. 4.
    E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968; Russian translation: Mir, Moscow, 1972.Google Scholar
  5. 5.
    J. L. Dornstetter, “On the equivalence between Berlekamp's and Euclid's algorithms,” IEEE Trans. Inform. Theory, IT-33 (1987), no. 3, 428–431.MathSciNetGoogle Scholar
  6. 6.
    Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa, “A method for solving key equation for decoding Goppa codes,” Inform. Control, 27 (1975), no. 1, 87–99.CrossRefMathSciNetGoogle Scholar
  7. 7.
    L. R. Welch and R. A. Scholtz, “Continued fractions and Berlekamp's algorithm,” IEEE Trans. Inform. Theory, IT-25 (1979), no. 1, 18–27.MathSciNetGoogle Scholar
  8. 8.
    U. Cheng, “On the continued fractions and Berlekamp's algorithm,” IEEE Trans. Inform. Theory, IT-30 (1984), no. 3, 541–544.Google Scholar
  9. 9.
    W. H. Mills, “Continued fractions and linear recurrence,” Math. Comp., 29 (1975), 173–180.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Zongduo Dai and Kencheng Zeng, “Continued fractions and Berlekamp—Massey algorithm,” in: Advances in Cryptology—Auscript-90, Springer-Verlag, Berlin, 1990, pp. 24–31.Google Scholar
  11. 11.
    E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality [in Russian] Nauka, Moscow, 1988.Google Scholar
  12. 12.
    G. Szego, Orthogonal Polynomials, Colloquium Publ., vol. XXIII, Amer. Math. Soc., Providence, RI, 1959; Russian translation, Fizmatgiz, Moscow, 1962.Google Scholar
  13. 13.
    V. M. Sidel'nikov, “Decoding of the Reed—Solomon codes with the number of errors greater than (d − 1)/2 and zeros of polynomials of several variables,” Problemy Peredachi Informatsii [Problems Inform. Transmission], 30 (1994), no. 1, 51–69.zbMATHMathSciNetGoogle Scholar
  14. 14.
    R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1994; Russian translation: Mir, Moscow, 1998.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. B. Gashkov
    • 1
  • I. B. Gashkov
    • 2
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Karlstads UniversitetSweden

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