Designs, Codes and Cryptography

, Volume 66, Issue 1–3, pp 3–16 | Cite as

Algebraic decoding of negacyclic codes over \({\mathbb Z_4}\)

  • Eimear Byrne
  • Marcus Greferath
  • Jaume Pernas
  • Jens ZumbrägelEmail author


In this article we investigate Berlekamp’s negacyclic codes and discover that these codes, when considered over the integers modulo 4, do not suffer any of the restrictions on the minimum distance observed in Berlekamp’s original papers: our codes have minimum Lee distance at least 2t + 1, where the generator polynomial of the code has roots α, α 3, . . . , α 2t-1 for a primitive 2nth root α of unity in a Galois extension of \({\mathbb {Z}_4}\) ; no restriction on t is imposed. We present an algebraic decoding algorithm for this class of codes that corrects any error pattern of Lee weight ≤ t. Our treatment uses Gröbner bases, the decoding complexity is quadratic in t.


Negacyclic code Integers modulo 4 Lee metric Galois ring Decoding Gröbner bases Key equation Solution by approximations Module of solutions 

Mathematics Subject Classification

94B15 94B35 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Eimear Byrne
    • 1
  • Marcus Greferath
    • 1
  • Jaume Pernas
    • 2
  • Jens Zumbrägel
    • 1
    Email author
  1. 1.Claude Shannon Institute, School of Mathematical SciencesUniversity College DublinDublin 4Ireland
  2. 2.Departament d’Enginyeria de la Informació i de les ComunicacionsUniversitat Autonoma de BarcelonaBellaterraSpain

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