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

TL;DR

This paper investigates the structure of $bZ_p bZ_{p^2}$-additive cyclic codes, focusing on their kernel and rank, and provides explicit generator polynomials and values for specific classes of these codes.

Contribution

It introduces the study of kernel and rank for $bZ_p bZ_{p^2}$-additive cyclic codes and determines their generator polynomials, advancing understanding of their algebraic structure.

Findings

Both the span and kernel codes are $bZ_p bZ_{p^2}$-additive cyclic codes.

Generator polynomials for these codes are explicitly determined.

Exact rank and kernel dimensions are computed for certain classes.

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.