Dimensions of the hull of generalized Reed-Solomon codes

Let GRS ${}_k(\mathit{\alpha },\mathit{\upsilon })$ be a $k$-dimensional generalized Reed-Solomon (GRS) code over ${\mathbb{F}}_q$ associated with $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_n)$ and $\mathit{\upsilon }=({\upsilon }_1,\dots ,{\upsilon }_n)$. In this paper, we determined the dimension of the Euclidean hull GRS ${}_k(\mathit{\alpha },\mathit{\upsilon })\cap$ GRS ${}_k(\mathit{\alpha },\mathit{\upsilon }{)}^{\mathrm{\perp }}$, which addresses an open problem posed in [Chen et al., IEEE-TIT, 2023]. We also presentd a new approach to generating all self-dual RS codes.