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

TL;DR

This paper establishes a link between vanishing Krein parameters in association schemes and the structure of designs, leading to new nonexistence results in finite geometry, including for generalized octagons and other geometrical structures.

Contribution

It proves that vanishing Krein parameters imply specific structural properties of designs, and applies this to derive new nonexistence results in finite geometry.

Findings

Nontrivial m-ovoids of generalized octagons of order (s, s^2) do not exist.

Simplified proofs for properties of partial geometries and generalized quadrangles.

New insights into the structure of association schemes and their applications in finite geometry.

Abstract

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 $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), 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 $(s, s^2)$ 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 $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.