Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

TL;DR

This paper investigates cyclic codes over a specific non-chain ring, introduces a Gray map to generate MDS and optimal LCD codes over finite fields, and establishes conditions for their existence and properties.

Contribution

It defines a Gray map for cyclic codes over a non-chain ring and constructs new MDS and optimal LCD codes with explicit conditions for their existence.

Findings

Constructed many MDS and optimal LCD codes from Gray images.

Established equivalence between LCD codes over the ring and their Gray images.

Provided conditions for the existence of free and non-free LCD codes.

Abstract

Let $\mathbb{F}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $t\geq 1,e\geq 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}=\mathbb{F}_q[u]/\langle u^e-1\rangle$. We define a Gray map $\varphi$ and obtain many { maximum-distance-separable} (MDS) and optimal $\mathbb{F}_q$-linear codes from the Gray images of cyclic codes. Under certain conditions we determine { linear complementary dual} (LCD) codes of length $n$ when $\gcd(n,q)\neq 1$ and $\gcd(n,q)= 1$, respectively. It is proved that { a} cyclic code $\mathcal{C}$ of length $n$ is an LCD code if and only if its Gray image $\varphi(\mathcal{C})$ is an LCD code of length $4n$ over $\mathbb{F}_q$. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over $R_{e,q}$.