${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$-additive codes

In this paper, we study ${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$-additive codes, where $p$ is prime and $u^2=0$. In particular, we determine a Gray map from ${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$ to ${\mathbb{Z}}_p^{\alpha +2\beta }$ and study generator and parity check matrices for these codes. We prove that a Gray map $\mathrm{\Phi }$ is a distance preserving map from (${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$,Gray distance) to (${\mathbb{Z}}_p^{\alpha +2\beta }$,Hamming distance), it is a weight preserving map as well. Furthermore we study the structure of ${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$-additive cyclic codes.