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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 Euclideanself-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two generalmethods to construct $e$-Galois self-orthogonal (extended) generalizedReed-Solomon (GRS) codes. As a consequence, eight new classes of $e$-Galoisself-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 shortenedcodes. As applications, new $e'$-Galois self-orthogonal maximum distanceseparable (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 codeswith prescribed dimensional $e$-Galois hull via the punctured codes arederived. Moreover, some new $\sqrt{q}$-ary quantum MDS codes with lengthsgreater than $\sqrt{q}+1$ and minimum distances greater than$\frac{\sqrt{q}}{2}+1$ are obtained.

Several classes of Galois self-orthogonal MDS codes and related applications