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

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 $\mathbb{Z}{p}[x]/\langle x^{{\alpha}-1\rangle} \times \mathbb{Z}{p}[u,v][x]/\langle x^{{\beta}-1\rangle$} 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.