  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 $\F_q$.
