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.