Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

In this paper we show that if θ is a T-design of an association scheme (Ω,ℛ), and the Krein parameters q i,j h vanish for some h∉T and all i,j∉T (i,j,h≠0), then θ consists of precisely half of the vertices of (Ω,ℛ) or it is a T ′ -design, where |T ′ |>|T|. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial m-ovoids of generalised octagons of order (s,s 2 ) do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order (s,s 2 ); (iii) the dual polar spaces DQ(2d,q), DW(2d-1,q) and DH(2d-1,q 2 ), for d≥3; (iv) the Penttila–Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila–Williford scheme in Q - (2n-1,q), n⩾3.