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

Abstract

Let ${\mathbb{F}}_q$ be a finite field of order $q$, a prime power integer such that $q=et+1$ where $t\ge 1,e\ge 2$ are integers. In this paper, we study cyclic codes of length $n$ over a non-chain ring $R_{e,q}={\mathbb{F}}_q[u]/⟨u^e-1⟩$. We define a Gray map $\phi$ 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 $\gcd (n,q)\ne 1$ 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 $\phi (\mathcal{C})$ 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 $R_{e,q}$.

Submitted to Discrete Mathematics