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

Abstract

Let \(\Delta\) be one of the dual polar spaces \(\mathrm{DQ}(8, q)\), \(\mathrm{DQ}^-(7,q)\), 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 \(\Delta \cong \mathrm{DQ}^-(7,q)\), then \(e(\Delta)\) is a \((q^3 + 1)\)-tight set of a nonsingular hyperbolic quadric \(\mathrm{Q}^+(7,q^2)\) of \(\Sigma \cong PG(7,q^2)\). 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.