Rank and pairs of Rank and Dimension of Kernel of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear codes

A code $C$ is called ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear if it is the Gray image of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code. For any prime number $p$ larger than $3$, the bounds of the rank of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear codes are given. For each value of the rank and the pairs of rank and the dimension of the kernel of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear codes, we give detailed construction of the corresponding codes. Finally, as an example, the rank and the dimension of the kernel of ${\mathbb{Z}}_5{\mathbb{Z}}_{25}$-linear codes are studied.