论文标题
FQ + UFQ + VFQ + UVFQ上的Galois LCD代码
Galois LCD Codes Over Fq + uFq + vFq + uvFq
论文作者
论文摘要
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 $e$.在这项工作中，我们将结果扩展到有限的非链环$ \ Mathcal {r} = \ Mathbb {f} _Q+U \ u \ Mathbb {f} _q+V \ Mathbb {f} _q+uv+uv \ uv \ uv \ uv \ uv \ mathb {f} _q $我们定义了$ l $ -galois在$ \ mathcal {r} $和$ l $ -galois上的线性代码之间的对应关系，其组件代码的双重代码是$ \ mathbb {f} _q。$ $ $ c。我们构建了euclidean lcd和$ l $ -galois lcd $ $ code $ code $ code $ code $ code。因此，这使我们证明了$ \ Mathcal {r} $上的任何线性代码相当于Euclidean（$ Q> 3 $）和$ l $ -galois lcd（$ 0 <l <e $，和$ p^{e-e-l} +1+1 \中间\中间\中\ \中的P^e-1 $而不是$ \ Mathcal $ cod $ cod。 \ Mathcal {r}。$
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} .$
