Let \(\mathscr {W}\) be a non-empty set of points of a finite Desarguesian projective space \(\textrm{PG}(n,q)\) . A collection of varieties of \(\textrm{PG}(n,q)\) is mutually \(\mu \) -intersecting (relatively to \(\mathscr {W}\) ) if its elements meet all \(\mathscr {W}\) in the same number of points and pairwise intersect in \(\mathscr {W}\) in exactly \(\mu \) points. Here, we construct a new family of mutually \(\mu \) -intersecting algebraic varieties by using certain quasi-Hermitian varieties of \(\textrm{PG}(n, q^2)\) , 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 \(\mu =q^{2n-3}\) .
