论文标题
几类Galois自动执行MDS代码和相关应用程序
Several classes of Galois self-orthogonal MDS codes and related applications
论文作者
论文摘要
令$ q = p^h $为主要功率，$ e $是$ 0 \ leq e \ leq h-1 $的整数。 $ e $ -galois自动守则是欧几里得自动守则（$ e = 0 $）和赫米尔人自动独立代码（$ e = \ frac {h} {2} $，$ h $的概括）。在本文中，我们提出了两种构建$ e $ galois自动式（扩展）广义的芦苇 - 溶解（GRS）代码的通用方法。结果，获得了八个新类别的$ e $ -galois自动式（扩展）GRS代码，其中奇数$ Q $和$ 2E \ MID H $。根据代码的Galois双重双重，我们还研究了其刺穿和缩短的代码。作为应用程序，新的$ e'$ - GALOIS自动距离最大距离可分开（MDS）代码，适用于满足$ 0 \ leq e'\ leq e'\ leq h-1 $，新$ e $ e $ -galois selfortortonal mds代码的新$ e'\ leq e'\ leq e'\ e'$，并通过缩短的代码和新的MDS代码，并带有新的MDS代码。此外，获得大于$ \ sqrt {q}+1 $的新$ \ sqrt {q} $ - ARY量子MDS代码，最小距离大于$ \ frac {\ frac {\ sqrt {q}}}} {2} {2} {2}+1 $。
Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two general methods to construct $e$-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of $e$-Galois self-orthogonal (extended) GRS codes with odd $q$ and $2e\mid h$ are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new $e'$-Galois self-orthogonal maximum distance separable (MDS) codes for all possible $e'$ satisfying $0\leq e'\leq h-1$, new $e$-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional $e$-Galois hull via the punctured codes are derived. Moreover, some new $\sqrt{q}$-ary quantum MDS codes with lengths greater than $\sqrt{q}+1$ and minimum distances greater than $\frac{\sqrt{q}}{2}+1$ are obtained.
