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

Abstract

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.