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

TL;DR

This paper investigates the properties of $Z_pZ_{p^2}$-linear codes, providing bounds on their rank, detailed constructions for various parameters, and an example analysis for codes over $Z_5Z_{25}$.

Contribution

It offers new bounds on the rank of $Z_pZ_{p^2}$-linear codes and detailed constructions for different rank and kernel dimensions, including a specific case study.

Findings

Bounds on the rank of $Z_pZ_{p^2}$-linear codes for $p > 3

Explicit constructions for codes with given rank and kernel dimension

Analysis of rank and kernel dimension for $Z_5Z_{25}$-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.