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

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.