Galois LCD Codes Over Fq + uFq + vFq + uvFq

TL;DR

This paper extends the study of Galois LCD codes from finite chain rings to a more complex non-chain ring, establishing duality correspondences, constructing LCD codes, and analyzing MDS codes over this ring.

Contribution

It generalizes Galois LCD code theory to a non-chain ring and constructs new LCD codes, providing a broader framework for code equivalence and properties.

Findings

Established a correspondence between Galois duals over the ring and its components.

Constructed Euclidean and Galois LCD codes from linear codes over the ring.

Proved that any code over the ring is equivalent to an LCD code under certain conditions.

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} .$