Spin-embeddings, two-intersection sets and two- weight codes

Let $\Delta$ be one of the dual polar spaces $DQ(8,q)$, $DQ^-(7,q)$, and let $e:\Delta \to \Sigma$ denote the spin-embedding of $\Delta$. We show that $e(\Delta)$ is a two-intersection set of the projective space $\Sigma$. Moreover, if $\Delta \cong DQ^-(7,q)$, then $e(\Delta)$ is a $(q^3+1)$-tight set of a nonsingular hyperbolic quadric $Q^+(7,q^2)$ of $\Sigma \cong \PG(7,q^2)$. This $(q^3+1)$-tight set gives rise to more examples of $(q^3+1)$-tight sets of hyperbolic quadrics by a procedure called field-reduction. All the above examples of two-intersection sets and $(q^3+1)$-tight sets give rise to two-weight codes and strongly regular graphs.