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 -> 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 congruent to DQ(-) (7, q), then e(Delta) is a (q(3) + 1)-tight set of a nonsingular hyperbolic quadric Q(+) (7, q(2)) of Sigma congruent to 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.

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