$\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes

TL;DR

This paper explores $ Z_p\nZ_p[u]$-additive codes, establishing a Gray map that preserves distance and weight, and analyzes their generator, parity check matrices, and cyclic code structures.

Contribution

It introduces a Gray map for $ Z_p\nZ_p[u]$-additive codes, proves its distance and weight preservation, and studies the structure of cyclic codes over this algebra.

Findings

Gray map is distance and weight preserving.

Characterization of generator and parity check matrices.

Structural analysis of cyclic codes over $ Z_p\nZ_p[u]$.

Abstract

In this paper, we study $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive codes, where $p$ is prime and $u^{2}=0$. In particular, we determine a Gray map from $ \mathbb{Z}_p\mathbb{Z}_p[u]$ to $\mathbb{Z}_p^{ \alpha+2 \beta}$ and study generator and parity check matrices for these codes. We prove that a Gray map $\Phi$ is a distance preserving map from ($\mathbb{Z}_p\mathbb{Z}_p[u]$,Gray distance) to ($\mathbb{Z}_p^{\alpha+2\beta}$,Hamming distance), it is a weight preserving map as well. Furthermore we study the structure of $\mathbb{Z}_p\mathbb{Z}_p[u]$-additive cyclic codes.