$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

  • Authors:

  • J. Borges;C. Fernández-Córdoba;J. Pujol;J. Rifà;M. Villanueva

  • Affiliations:

  • Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, Bellaterra, Spain 08193;Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, Bellaterra, Spain 08193;Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, Bellaterra, Spain 08193;Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, Bellaterra, Spain 08193;Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, Bellaterra, Spain 08193

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2010

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Abstract

A code $${{\mathcal C}}$$ is $${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of $${{\mathcal C}}$$ by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). In this paper $${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -additive codes are studied. Their corresponding binary images, via the Gray map, are $${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for $${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -additive codes is defined and the parameters of their dual codes are computed.