Gray Images of Cyclic Codes over $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p\mathbb{Z}_{p^2}

TL;DR

This paper investigates the algebraic structure of cyclic codes over mixed rings and characterizes when their Gray images are linear, providing generator polynomials and conditions for linearity.

Contribution

It introduces the algebraic framework for $ ext{Z}_p ext{Z}_{p^k}$-additive cyclic codes and establishes criteria for the linearity of their Gray images.

Findings

Generator polynomials for these codes are derived.

Necessary and sufficient conditions for Gray image linearity are provided.

Linearity of Gray images for specific codes over Z_9 and Z_3Z_9 is determined.

Abstract

In the paper, we firstly study the algebraic structures of $\mathbb{Z}_p \mathbb{Z}_{p^k}$-additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for a class of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes whose Gray images are linear (not necessarily cyclic) over $\mathbb{Z}_p$ is given. Moreover, as for some special families of cyclic codes over $\mathbb{Z}_{9}$ and $\mathbb{Z}_3 \mathbb{Z}_{9}$, the linearity of the Gray images is determined.