Computing the generator polynomials of ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic codes

Abstract

A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code $\mathcal{C}\subseteq {\mathbb{Z}}_2^{\alpha }\times {\mathbb{Z}}_4^{\beta }$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any simultaneous cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the ${\mathbb{Z}}_4[x]$-module ${\mathbb{Z}}_2[x]/(x^{\alpha }-1)\times {\mathbb{Z}}_4[x]/(x^{\beta }-1)$. Any ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code $\mathcal{C}$ is of the form $⟨(b(x)\mid 0),(\ell (x)\mid f(x)h(x)+2f(x))⟩$ for some $b(x),\ell (x)\in {\mathbb{Z}}_2[x]/(x^{\alpha }-1)$ and $f(x),h(x)\in {\mathbb{Z}}_4[x]/(x^{\beta }-1)$. A new algorithm is presented to compute the generator polynomials for ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic codes.