Constructions of tight sets of the Hermitian polar space $\mc{H}(2r-1,q^2)$

TL;DR

This paper constructs two infinite families of tight sets in Hermitian polar spaces, expanding the understanding of their combinatorial and geometric structures with specified automorphism groups.

Contribution

It introduces new infinite families of tight sets in Hermitian polar spaces with explicit parameters and automorphism groups.

Findings

Constructed two infinite families of tight sets in $ ext{H}(2r-1,q^2)$.

Determined the automorphism groups of these tight sets.

Provided parameters for the tight sets applicable for all $r \\ge 2$ and prime powers $q$.

Abstract

In this paper, we construct two infinite families of tight sets with parameters $(q^{2r-2}-1)$ and $(q^{2r-1}-q^{2r-2})$, respectively, in the Hermitian polar space $\mathcal{H}(2r-1,q^2)$ for any $r\ge 2$ and any prime power $q$. Both families admit $(q-1).\PGL(r,q^2).2.2e$ as the full automorphism group, where $q=p^e$, $p$ is a prime, and $e$ a positive integer.