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 $(Ω, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $θ$ consists of precisely half of the vertices of $(Ω, \mathcal{R})$ 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 $\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$; (iv) the Penttila-Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.

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