Intersections of linear codes and related MDS codes with new Galois hulls

Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. In this paper, we investigate general properties of linear codes associated with $\sigma$ duals for $\sigma\in\mathrm{SLAut}(\mathbb{F}_{q}^{n})$. We show that the dimension of the intersection of two linear codes can be determined by generator matrices of such codes and their $\sigma$ duals. We also show that the dimension of $\sigma$ hull of a linear code can be determined by a generator matrix of it or its $\sigma$ dual. We give a characterization on $\sigma$ dual and $\sigma$ hull of a matrix-product code. We also investigate the intersection of a pair of matrix-product codes. We provide a necessary and sufficient condition under which any codeword of a generalized Reed-Solomon (GRS) code or an extended GRS code is contained in its $\sigma$ dual. As an application, we construct eleven families of $q$-ary MDS codes with new $\ell$-Galois hulls satisfying $2(e-\ell)\mid e$, which are not covered by the latest papers by Cao (IEEE Trans. Inf. Theory 67(12), 7964-7984, 2021) and by Fang et al. (Cryptogr. Commun. 14(1), 145-159, 2022) when $\ell\neq \frac{e}{2}$.