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

Minjia Shi; Shukai Wang; Xiaoxiao Li

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arXiv · 2205.13981

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

Minjia Shi·Shukai Wang·Xiaoxiao Li

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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.

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Minjia Shi, Shukai Wang, Xiaoxiao Li. 2022-05-27. $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear codes: rank and kernel. https://arxiv.org/abs/2205.13981

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