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.

翻译：设$q=p^h$为素数幂，$e$为满足$0\leq e\leq h-1$的整数。$e$-伽罗瓦自对偶码是欧几里得自对偶码（$e=0$）与埃尔米特自对偶码（$e=\frac{h}{2}$且$h$为偶数）的推广。本文提出两种构造$e$-伽罗瓦自对偶（扩展）广义里德-所罗门码的通用方法。由此，在$q$为奇数且$2e\mid h$的条件下，得到八类新的$e$-伽罗瓦自对偶（扩展）广义里德-所罗门码。基于码的伽罗瓦对偶，我们还研究了其删余码与缩短码。作为应用，推导出：对所有满足$0\leq e'\leq h-1$的$e'$，新型$e'$-伽罗瓦自对偶最大距离可分码；通过缩短码得到的新型$e$-伽罗瓦自对偶最大距离可分码；以及通过删余码得到的具有指定维数$e$-伽罗瓦核的最大距离可分码。此外，还得到若干长度大于$\sqrt{q}+1$且最小距离大于$\frac{\sqrt{q}}{2}+1$的新型$\sqrt{q}$元量子最大距离可分码。