A new family of $(q^4+1)$-tight sets with an automorphism group $F_4(q)$

TL;DR

This paper introduces a novel construction of tight sets in certain quadrics using the action of the exceptional group F_4(q), expanding the understanding of geometric structures linked to algebraic groups.

Contribution

It presents a new family of tight sets in quadrics derived from the action of the exceptional group F_4(q), utilizing its minimal module over finite fields.

Findings

Construction of $(q^4+1)$-tight sets in specific quadrics.

Use of the group $F_4(q)$ action on its minimal module.

Applicable for $q=3^f$ or $q\equiv 2\pmod 3$.

Abstract

In this paper, we construct a new family of $(q^4+1)$-tight sets in $Q(24,q)$ or $Q^-(25,q)$ according as $q=3^f$ or $q\equiv 2\pmod 3$. The novelty of the construction is the use of the action of the exceptional simple group $F_4(q)$ on its minimal module over $\F_q$.