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

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 $q\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$. The novelty of the construction is the use of the action of the exceptional simple group $F_4(q)$ on its minimal module over ${\text{F}}_q$.