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

Abstract

A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ 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 ${\cal C}$ is of the form $\langle (b(x)\mid{ 0}), (\ell(x) \mid f(x)h(x) +2f(x)) \rangle$ 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.