Designs, Codes and Cryptography

, Volume 56, Issue 1, pp 43–59

\({\mathbb{Z}_2\mathbb{Z}_4}\)-linear codes: rank and kernel

  • Cristina Fernández-Córdoba
  • Jaume Pujol
  • Mercè Villanueva
Article

DOI: 10.1007/s10623-009-9340-9

Cite this article as:
Fernández-Córdoba, C., Pujol, J. & Villanueva, M. Des. Codes Cryptogr. (2010) 56: 43. doi:10.1007/s10623-009-9340-9

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). The corresponding binary codes of \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-additive codes under an extended Gray map are called \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear codes. In this paper, the invariants for \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear code for each possible pair (r, k) is given.

Keywords

Quaternary linear codes \({\mathbb{Z}_4}\)-linear codes \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-additive codes \({{\mathbb{Z}_2\mathbb{Z}_4}}\)-linear codes Kernel Rank 

Mathematics Subject Classification (2000)

94B60 94B25 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Cristina Fernández-Córdoba
    • 1
  • Jaume Pujol
    • 1
  • Mercè Villanueva
    • 1
  1. 1.Department of Information and Communications EngineeringUniversitat Autònoma de BarcelonaBellaterraSpain