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From the Euclidean Algorithm for Solving a Key Equation for Dual Reed–Solomon Codes to the Berlekamp–Massey Algorithm

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5527))

Abstract

The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. This article presents a new version of the key equation and a way to use the Euclidean algorithm to solve it. A straightforward reorganization of the algorithm yields the Berlekamp-Massey algorithm.

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References

  1. Berlekamp, E.R.: Algebraic coding theory. McGraw-Hill Book Co., New York (1968)

    MATH  Google Scholar 

  2. Cox, D.A., Little, J., O’Shea, D. (eds.): Using algebraic geometry, 2nd edn. Graduate Texts in Mathematics, vol. 185. Springer, New York (2005)

    MATH  Google Scholar 

  3. Dornstetter, J.-L.: On the equivalence between Berlekamp’s and Euclid’s algorithms. IEEE Trans. Inform. Theory 33(3), 428–431 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. David Forney Jr., G.: On decoding BCH codes. IEEE Trans. Information Theory IT-11, 549–557 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  5. Heydtmann, A.E., Jensen, J.M.: On the equivalence of the Berlekamp-Massey and the Euclidean algorithms for decoding. IEEE Trans. Inform. Theory 46(7), 2614–2624 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Information Theory IT-15, 122–127 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. O’Sullivan, M.E., Bras-Amorós, M.: The Key Equation for One-Point Codes, ch. 3, pp. 99–152 (2008)

    Google Scholar 

  8. Roth, R.: Introduction to coding theory. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  9. Sugiyama, Y., Kasahara, M., Hirasawa, S., Namekawa, T.: A method for solving key equation for decoding Goppa codes. Information and Control 27, 87–99 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Spain

    Maria Bras-Amorós

  2. Department of Mathematics and Statistics, San Diego State University, USA

    Michael E. O’Sullivan

Authors
  1. Maria Bras-Amorós
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  2. Michael E. O’Sullivan
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Editor information

Editors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avinguda Països Catalans, 26, 43007, Tarragona, Spain

    Maria Bras-Amorós

  2. Department of Mathematics, The Technical University of Denmark, Building 303, 2800, Lyngby, Denmark

    Tom Høholdt

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© 2009 Springer-Verlag Berlin Heidelberg

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Bras-Amorós, M., O’Sullivan, M.E. (2009). From the Euclidean Algorithm for Solving a Key Equation for Dual Reed–Solomon Codes to the Berlekamp–Massey Algorithm. In: Bras-Amorós, M., Høholdt, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2009. Lecture Notes in Computer Science, vol 5527. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02181-7_4

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  • DOI: https://doi.org/10.1007/978-3-642-02181-7_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02180-0

  • Online ISBN: 978-3-642-02181-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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