Galois LCD Codes Over Fq + uFq + vFq + uvFq

Abstract

In \cite{anote}, Wu and Shi studied $l$-Galois LCD codes over finite chain ring $\mathcal{R}={\mathbb{F}}_q+u{\mathbb{F}}_q$, where $u^2=0$ and $q=p^e$ for some prime $p$ and positive integer $e$. In this work, we extend the results to the finite non chain ring $\mathcal{R}={\mathbb{F}}_q+u{\mathbb{F}}_q+v{\mathbb{F}}_q+uv{\mathbb{F}}_q$, where $u^2=u,v^2=v$ and $uv=vu$. We define a correspondence between $l$-Galois dual of linear codes over $\mathcal{R}$ and $l$-Galois dual of its component codes over ${\mathbb{F}}_q.$ Further, we construct Euclidean LCD and $l$-Galois LCD codes from linear code over $\mathcal{R}$. This consequently leads us to prove that any linear code over $\mathcal{R}$ is equivalent to Euclidean ($q>3$) and $l$-Galois LCD ($0<l<e$, and $p^{e-l}+1\mid p^e-1$) code over $\mathcal{R}.$ Finally, we investigate MDS codes over $\mathcal{R}.$