Skip to main content
SpringerLink
Log in

Refined convergents to the associated continued fractions for binary sequences

  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript

Abstract

The relation between continued fractions and Berlekamp's algorithm was studied by some researchers. The latter is an iterative procedure proposed for decoding BCH codes. However, there remains an unanswered question wheter each of the iterative steps in the algorithm can be interpreted in terms of continued fractions. In this paper, we first introduce the so-called refined convergents to the continued fraction expansion of a binary sequence S, and then give a thorough answer to the question in the context of Massey's linear feedback shift register synthesis algorithm which is equivalent to that of Berlekamp, and at last we prove that there exists a one-to-one correspondence between then-th refined convergents and the lengthn segments.

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. Beker, H. and Piper, F., Cipher System, Northwood Books, London, 1982.

    Google Scholar 

  2. Rueppel, R.A., Analysis Design of Stream Cipher, Springer, Berlin-Heidelberg-New York-London-Paris-Tokyo: Springer-Verlag. 1986.

    Google Scholar 

  3. Berlekamp, E.R., Algebraic Coding Theory, New York: MacGraw-Hill, 1968.

    MATH  Google Scholar 

  4. Massey, J.L.,Shift-register synthesis and BCH decoding, IEEE Trans. Info. Th.,15(1969), 122–127.

    Article  MATH  MathSciNet  Google Scholar 

  5. Sugiyama, Y., M. Kaschara, S. Hirasawa and T. Namekawa,A method for solving key equation for decoding goppa codes, Info. and Control,29(1975), 173–180.

    Google Scholar 

  6. Mills, W.H.,Continued fractions and linear recurrences, Math. Comp.,29(1975), 173–180.

    Article  MATH  MathSciNet  Google Scholar 

  7. Dai Z.D., & Wan, Z.X.,A relationship between the Berlekamp-Massey and the Euclidean algorithms for linear feedback shift register synthesis, Acta Mathematics Sinica, New Series,4(1988), 55–63.

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng, U.,On the continued fraction and Berlekamp's Algorithm, IEEE Trans. Info. Th.,30(1984), 541–544.

    Article  MATH  Google Scholar 

  9. Reed, I.S., Scholtz, R.A., Truong, T.K. and Welch, I.R.,The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions, IEEE Trans. Information Theory,24(1978), 100–106.

    Article  MATH  MathSciNet  Google Scholar 

  10. Welch, L.R. and Scholtz, R.A.,Continued fraction and Berlekamp's algorithm, IEEE Trans. Information Theory,25(1979), 19–27.

    Article  MATH  MathSciNet  Google Scholar 

  11. Wan, Z., Algebra and Coding Theory, Science Publishing House, Beijing, 1976.

    Google Scholar 

  12. Devenport, H., Higher Arithmetic, 5th ed., Cambridge University Press, 1982.

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Security, Academia Sinica, 100039, Beijing, China

    Dai Zongduo & Zeng Kencheng

Authors
  1. Dai Zongduo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zeng Kencheng
    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

Zongduo, D., Kencheng, Z. Refined convergents to the associated continued fractions for binary sequences. Acta Mathematica Sinica 10, 179–191 (1994). https://doi.org/10.1007/BF02580425

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02580425

Keywords

Search

Navigation