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, s²) 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, s²); (iii) the dual polar spaces DQ(2d, q), DW(2d - 1, q) and DH(2d-1, q²), 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.