Several classes of Galois self-orthogonal MDS codes

Let $q=p^h$ be an odd 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 of constructing several classes of $e$-Galois self-orthogonal generalized Reed-Solomn codes and extended generalized Reed-Solomn codes with $2e\mid h$. We can determine all possible $e$-Galois self-orthogonal maximum distance separable codes of certain lengths for each even $h$ and odd prime number $p$.