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

TL;DR

This paper introduces a new family of mutually $oldsymbol{ extmu}$-intersecting algebraic varieties in finite projective spaces, leveraging quasi-Hermitian varieties to construct optimal codes and arrays with potential applications in combinatorics and coding theory.

Contribution

It constructs a novel family of mutually $oldsymbol{ extmu}$-intersecting varieties using quasi-Hermitian varieties, enabling new code and array constructions.

Findings

Constructed a new family of mutually $ extmu$-intersecting varieties.

Provided a new construction of 5-dimensional MDS codes over $oldsymbol{ extbf{F}_q}$.

Developed an infinite family of simple orthogonal arrays with specific parameters.

Abstract

Let $\mathcal W$ be a non-empty set of points of a finite Desarguesian projective space $\mathrm{PG}(n,q)$. A collection of varieties of $\mathrm{PG}(n,q)$ 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 $\mathrm{PG}(n,q^2)$ 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 $\mu=q^{2n-3}$.