Designs, Codes and Cryptography

, Volume 54, Issue 2, pp 167–179

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

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

DOI: 10.1007/s10623-009-9316-9

Cite this article as:
Borges, J., Fernández-Córdoba, C., Pujol, J. et al. Des. Codes Cryptogr. (2010) 54: 167. doi:10.1007/s10623-009-9316-9

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.

Mathematics Subject Classification (2000)

94B60 94B25 

Keywords

Binary linear codes Duality Quaternary linear codes \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -additive codes \({{{\mathbb Z}_2}{{\mathbb Z}_4}}\) -linear codes 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • J. Borges
    • 1
  • C. Fernández-Córdoba
    • 1
  • J. Pujol
    • 1
  • J. Rifà
    • 1
  • M. Villanueva
    • 1
  1. 1.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaBellaterraSpain