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

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, where $p>2$ is prime. In this paper, the rank and the dimension of the kernel of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear codes are studied. Two bounds of the rank of a ${\mathbb{Z}}_3{\mathbb{Z}}_9$-linear code and the dimension of the kernel of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear code are given, respectively. For each value of these bounds, we give detailed construction of the corresponding code. Finally, pairs of rank and the dimension of the kernel of ${\mathbb{Z}}_3{\mathbb{Z}}_9$-linear codes are also considered.