# Linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$

## TL;DR

This paper studies linear codes over a complex ring extension of our, analyzing their structure, weight enumerators, bounds, and special subclasses like cyclic codes, with new definitions and relations.

## Contribution

It introduces the structure of linear codes over a multi-component ring extension of our, including weight, Gray map, and MacWilliams relations, and explores cyclic and quasi-cyclic codes.

## Key findings

- Defined Lee weight and Gray map for these codes
- Derived MacWilliams relations for various weight enumerators
- Discussed bounds and examples of cyclic codes

## Abstract

We investigate linear codes over the ring $\mathbb{Z}_4 + u\mathbb{Z}_4 + v\mathbb{Z}_4 + w\mathbb{Z}_4 + uv\mathbb{Z}_4 + uw\mathbb{Z}_4 + vw\mathbb{Z}_4 + uvw\mathbb{Z}_4$, with conditions $u^2=u$, $v^2=v$, $w^2=w$, $uv=vu$, $uw=wu$ and $vw=wv.$ We first analyze the structure of the ring and then define linear codes over this ring. Lee weight and Gray map for these codes are defined and MacWilliams relations for complete, symmetrized, and Lee weight enumerators are obtained. The Singleton bound as well as maximum distance separable codes are also considered. Furthermore, cyclic and quasi-cyclic codes are discussed, and some examples are also provided.