Galois LCD Codes Over Fq + uFq + vFq + uvFq
(2206.08725)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 $u2=0$ and $ q=pe$ 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 $u2=u,v2=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 $p{e-l}+1\mid pe-1$) code over $ \mathcal{R} .$ Finally, we investigate MDS codes over $ \mathcal{R} .$
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