Spin-Embeddings, Two-Intersection Sets and Two-Weight Codes

Abstract

Let $\mathrm{\Delta }$ be one of the dual polar spaces $\mathrm{D}\mathrm{Q}(8,q)$, ${\mathrm{D}\mathrm{Q}}^{-}(7,q)$, and let $e:\mathrm{\Delta }\to \mathrm{\Sigma }$ denote the spin-embedding of $\mathrm{\Delta }$. We show that $e(\mathrm{\Delta })$ is a two-intersection set of the projective space $\mathrm{\Sigma }$. Moreover, if $\mathrm{\Delta }\cong {\mathrm{D}\mathrm{Q}}^{-}(7,q)$, then $e(\mathrm{\Delta })$ is a $(q^3+1)$-tight set of a nonsingular hyperbolic quadric ${\mathrm{Q}}^{+}(7,q^2)$ of $\mathrm{\Sigma }\cong PG(7,q^2)$. This $(q^2+1)$-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.