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

Xuan Wang; Minjia Shi

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arXiv · 2206.14201

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

Xuan Wang·Minjia Shi

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Abstract

A code $C = \Phi(\mathcal{C})$ is called $\mathbb{Z}_p \mathbb{Z}_{p^2}$-linear if it's the Gray image of the $\mathbb{Z}_p \mathbb{Z}_{p^2}$-additive code $\mathcal{C}$. In this paper, the rank and the dimension of the kernel of $\mathcal{C}$ are studied. Both of the codes $\langle \Phi(\mathcal{C}) \rangle$ and $\ker(\Phi(\mathcal{C}))$ are proven $\mathbb{Z}_p \mathbb{Z}_{p^2}$-additive cyclic codes, and their generator polynomials are determined. Finally, accurate values of rank and the dimension of the kernel of some classes of $\mathbb{Z}_p \mathbb{Z}_{p^2}$-additive cyclic codes are considered.

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Xuan Wang, Minjia Shi. 2022-06-28. $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive cyclic codes: kernel and rank. https://arxiv.org/abs/2206.14201

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