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

Abstract: 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.