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

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.