On mutually $\mu$-intersecting quasi-Hermitian varieties with some applications

Let $\mathcal{W}$ be a non-empty set of points of a finite Desarguesian projective space $\mathrm{P}\mathrm{G}(n,q)$. A collection of varieties of $\mathrm{P}\mathrm{G}(n,q)$ is \emph{mutually $\mu$-intersecting (relatively to $\mathcal{W}$)} if its elements meet all $\mathcal{W}$ in the same number of points and pairwise intersect in $\mathcal{W}$ in exactly $\mu$-points. Here we construct a new family of mutually $\mu$-intersecting algebraic varieties by using certain quasi-Hermitian varieties of $\mathrm{P}\mathrm{G}(n,q^2)$ where $q$ is any prime power. With the help of these quasi-Hermitian varieties we provide a new construction of $5$-dimensional MDS codes over ${\mathbb{F}}_q$ as well as an infinite family of simple orthogonal arrays $OA(q^{2n-1},q^{2n-2},q,2)$ of index $\mu =q^{2n-3}$.