On ZpZp[u, v]-additive cyclic and constacyclic codes

Abstract

Let ${\mathbb{Z}}_p$ be the ring of residue classes modulo a prime $p$. The ${\mathbb{Z}}_p{\mathbb{Z}}_p[u,v]$-additive cyclic codes of length $(\alpha ,\beta )$ is identify as ${\mathbb{Z}}_p[u,v][x]$-submodule of ${\mathbb{Z}}_p[x]/⟨x^{\alpha }-1⟩\times {\mathbb{Z}}_p[u,v][x]/⟨x^{\beta }-1⟩$ where ${\mathbb{Z}}_p[u,v]={\mathbb{Z}}_p+u{\mathbb{Z}}_p+v{\mathbb{Z}}_p$ with $u^2=v^2=uv=vu=0$. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length $\beta$ over ${\mathbb{Z}}_p[u,v]$ and ${\mathbb{Z}}_p{\mathbb{Z}}_p[u,v]$-additive cyclic codes with length $(\alpha ,\beta )$ respectively. We show that the Gray image of ${\mathbb{Z}}_p{\mathbb{Z}}_p[u,v]$-additive cyclic code with length $(\alpha ,\beta )$ is either a QC code of length $4\alpha$ with index $4$ or a generalized QC code of length $(\alpha ,3\beta )$ over ${\mathbb{Z}}_p$. Moreover, some structural properties like generating polynomials, minimal generating sets of ${\mathbb{Z}}_p{\mathbb{Z}}_p[u,v]$-additive constacyclic code with length $(\alpha ,p-1)$ are determined.

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