$(1+2u)$-constacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$

TL;DR

This paper introduces a new Gray map for a specific ring extension and studies constacyclic codes over it, showing their Gray images are cyclic over 04 and deriving new linear codes over 04.

Contribution

It presents a novel Gray map for 04+u04 and analyzes 01+2u04-constacyclic codes over this ring, linking them to cyclic codes over 04 and constructing new codes.

Findings

Gray image of constacyclic codes are cyclic over 04

Constructed new linear codes over 04 from these constacyclic codes

Established a correspondence between codes over ring and over 04

Abstract

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$. Further, a number of linear codes over $\mathbb{Z}_4$ as the images of $(1+2u)$-constacyclic codes over $R$ are obtained.