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

TL;DR

This paper investigates the properties of $ ext{Z}_p ext{Z}_{p^2}$-linear codes, focusing on their rank and kernel dimension, providing bounds, constructions, and analysis for specific cases like $ ext{Z}_3 ext{Z}_9$.

Contribution

It introduces bounds and constructions for the rank and kernel dimension of $ ext{Z}_p ext{Z}_{p^2}$-linear codes, expanding understanding of their structural properties.

Findings

Established bounds for the rank of $ ext{Z}_3 ext{Z}_9$-linear codes.

Determined bounds for the kernel dimension of $ ext{Z}_p ext{Z}_{p^2}$-linear codes.

Provided explicit constructions for codes achieving these bounds.

Abstract

A code $C$ is called $\Z_p\Z_{p^2}$-linear if it is the Gray image of a $\Z_p\Z_{p^2}$-additive code, where $p>2$ is prime. In this paper, the rank and the dimension of the kernel of $\Z_p\Z_{p^2}$-linear codes are studied. Two bounds of the rank of a $\Z_3\Z_{9}$-linear code and the dimension of the kernel of a $\Z_p\Z_{p^2}$-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 $\Z_3\Z_{9}$-linear codes are also considered.