Gray Images of Cyclic Codes over ${\mathbb{Z}}_{p^2}$ and $$\mathbb{Z}$_p$\mathbb{Z}$_{p^2}

In the paper, we firstly study the algebraic structures of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^k}$-additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for a class of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive codes whose Gray images are linear (not necessarily cyclic) over ${\mathbb{Z}}_p$ is given. Moreover, as for some special families of cyclic codes over ${\mathbb{Z}}_9$ and ${\mathbb{Z}}_3{\mathbb{Z}}_9$, the linearity of the Gray images is determined.