Z2Z4-additive cyclic codes, generator polynomials and dual 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 cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the ${\mathbb{Z}}_4[x]$-module ${\mathbb{Z}}_2[x]/(x^{\alpha }-1)\times {\mathbb{Z}}_4[x]/(x^{\beta }-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are determined in terms of the generator polynomials of the code $\mathcal{C}$.

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