On PGZ decoding of alternant codes

  • Rafel Farré
  • Narcís Sayols
  • Sebastià Xambó-DescampsEmail author


In this paper, we first review the classical Petterson–Gorenstein–Zierler decoding algorithm for the class of alternant codes, which includes Reed–Solomon, Bose–Chaudhuri–Hocquenghem and classical Goppa codes. Afterwards, we present an improvement of the method to find the number of errors and the error-locator polynomial. Finally, we illustrate the procedure with several examples. In two appendices, we sketch the main features of the computer algebra system designed and developed to support the computations.


Alternant codes RS codes BCH codes Classical Goppa codes Computer algebra systems 

Mathematics Subject Classification

11T71 94B05 94B35 94B15 



The authors are grateful to the anonymous referees for having pointed out a number of misprints and suggested some style changes.

Supplementary material


  1. Farré R (2003) Notes on information theory and coding theory. In CatalanGoogle Scholar
  2. Peterson W W, Weldon E J (1972) Error-correcting codes, 2nd edn. MIT Press, CambridgezbMATHGoogle Scholar
  3. Sayols N, Xambó-Descamps S (2017) A Python package for the construction, coding and decoding of error-correcting codes.
  4. Xambó-Descamps S (2003) Block error-correcting codes: a computational primer. Univesitext. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  • Rafel Farré
    • 1
  • Narcís Sayols
    • 1
  • Sebastià Xambó-Descamps
    • 1
    Email author
  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

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