Designs, Codes and Cryptography

, Volume 76, Issue 2, pp 269–277 | Cite as

Permutation decoding of \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-linear codes

  • José Joaquín Bernal
  • Joaquim BorgesEmail author
  • Cristina Fernández-Córdoba
  • Mercè Villanueva


An alternative permutation decoding method is described which can be used for any binary systematic encoding scheme, regardless whether the code is linear or not. Thus, the method can be applied to some important codes such as \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-linear codes, which are binary and, in general, nonlinear codes in the usual sense. For this, it is proved that these codes allow a systematic encoding scheme. As particular examples, this permutation decoding method is applied to some Hadamard \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-linear codes.


Permutation decoding \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-linear codes Hadamard codes 

Mathematics Subject Classification

94B60 94B25 



The authors thank Prof. J. Rifà for valuable discussions in an earlier version of this paper. They also thank the anonymous referees for their valuable comments, which enabled them to improve the quality of the paper. This work was partially supported by the Spanish MICINN under Grants TIN2010-17358 and TIN2013-40524-P, and by the Catalan AGAUR under Grant 2009SGR1224. The authors are in alphabetical order.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • José Joaquín Bernal
    • 1
  • Joaquim Borges
    • 2
    Email author
  • Cristina Fernández-Córdoba
    • 2
  • Mercè Villanueva
    • 2
  1. 1.Department of MathematicsUniversidad de MurciaEspinardoSpain
  2. 2.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaCerdanyola del VallèsSpain

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