In this paper we show that if $theta$ is a $T$-design of an association scheme $(Omega, mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h in T$ and all $i, j in T$, then $theta$ consists of precisely half of the vertices of $(Omega, mathcal{R})$ 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 $(s, s^2)$ are hemisystems, and hence no $m$-ovoid of a Ree-Tits octagon can 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 $rm{DQ}(2d, q)$, $rm{DW}(2d-1,q)$ and $rm{DH}(2d-1,q^2)$, 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 $rm{Q}^-(2n-1, q)$, $ngeqslant 3$.
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