Let $q=p^m$ be a prime power, $e$ be an integer with $0\leq e\leq m-1$ and $s=\gcd(e,m)$. In this paper, for a vector $v$ and a $q$-ary linear code $C$, we give some necessary and sufficient conditions for the equivalent code $vC$ of $C$ and the extended code of $vC$ to be $e$-Galois self-orthogonal. From this, we directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be $e$-Galois self-orthogonal. Furthermore, for all possible $e$ satisfying $0\leq e\leq m-1$, we classify them into three cases (1) $\frac{m}{s}$ odd and $p$ even; (2) $\frac{m}{s}$ odd and $p$ odd; (3) $\frac{m}{s}$ even, and construct several new classes of $e$-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our $e$-Galois self-orthogonal MDS codes can have dimensions greater than $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$, which are not covered by previously known ones. Moreover, by propagation rules, we obtain some new MDS codes with Galois hulls of arbitrary dimensions. As an application, many quantum codes can be obtained from these MDS codes with Galois hulls.

翻译：设 $q=p^m$ 为素数幂，$e$ 为满足 $0\leq e\leq m-1$ 的整数，且 $s=\gcd(e,m)$。本文针对向量 $v$ 和 $q$ 元线性码 $C$，给出了 $C$ 的等价码 $vC$ 及其扩展码为 $e$-Galois 自正交的一些充分必要条件。由此，我们直接得到了（扩展）广义Reed-Solomon码（GRS和EGRS码）为 $e$-Galois 自正交码的一些充分必要条件。进一步，对于所有满足 $0\leq e\leq m-1$ 的 $e$，我们将其分为三种情形：（1）$\frac{m}{s}$ 为奇数且 $p$ 为偶数；（2）$\frac{m}{s}$ 为奇数且 $p$ 为奇数；（3）$\frac{m}{s}$ 为偶数，并构造了若干类新的 $e$-Galois 自正交极大距离可分（MDS）码。值得注意的是，我们构造的 $e$-Galois 自正交 MDS 码的维数可以大于 $\lfloor \frac{n+p^e-1}{p^e+1}\rfloor$，这是以往已知结果未能覆盖的。此外，通过传播规则，我们得到了一些具有任意维数Galois壳的新MDS码。作为应用，可以从这些具有Galois壳的MDS码构造出许多量子码。