  Mutually intersecting varieties of a finite Desarguesian projective space are
varieties with the same size, such that the intersection of two of them
contains the same number of points. Here, we show the existence of a new family
of mutually intersecting varieties by using certain quasi-Hermitian varieties
of $\mathrm{PG}(n,q^2)$, 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 $\mu=q^{2n-3}$.
