${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive cyclic codes: kernel and rank

A code $C=\mathrm{\Phi }(\mathcal{C})$ is called ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear if it’s the Gray image of the ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code $\mathcal{C}$. In this paper, the rank and the dimension of the kernel of $\mathcal{C}$ are studied. Both of the codes $⟨\mathrm{\Phi }(\mathcal{C})⟩$ and $\ker (\mathrm{\Phi }(\mathcal{C}))$ are proven ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive cyclic codes, and their generator polynomials are determined. Finally, accurate values of rank and the dimension of the kernel of some classes of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive cyclic codes are considered.