# ALGEBRAIC COMBINATORICS

Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 197-212.

In this paper we show that if $\theta$ is a $T$-design of an association scheme $\left(\Omega ,ℛ\right)$, and the Krein parameters ${q}_{i,j}^{h}$ vanish for some $h\notin T$ and all $i,j\notin T$ ($i,j,h\ne 0$), then $\theta$ consists of precisely half of the vertices of $\left(\Omega ,ℛ\right)$ or it is a ${T}^{\prime }$-design, where $|{T}^{\prime }|>|T|$. 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 $\left(s,{s}^{2}\right)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $\left(s,{s}^{2}\right)$; (iii) the dual polar spaces $\mathsf{DQ}\left(2d,q\right)$, $\mathsf{DW}\left(2d-1,q\right)$ and $\mathsf{DH}\left(2d-1,{q}^{2}\right)$, 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 ${\mathsf{Q}}^{-}\left(2n-1,q\right)$, $n⩾3$.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.246
Classification: 05E30, 05B30, 05B25, 51E12, 51E05
Keywords: association schemes, Delsarte designs, Krein parameters, hemisystems, $m$-ovoids, generalised octagons, finite geometry
Bamberg, John 1; Lansdown, Jesse 1

1 Centre for the Mathematics of Symmetry and Computation Department of Mathematics and Statistics The University of Western Australia, W.A., Australia
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Bamberg, John; Lansdown, Jesse. Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 197-212. doi : 10.5802/alco.246. https://alco.centre-mersenne.org/articles/10.5802/alco.246/

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