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

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