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 θ is a T-design of an association scheme (Ω,), and the Krein parameters q i,j h vanish for some hT and all i,jT (i,j,h0), then θ consists of precisely half of the vertices of (Ω,) 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 DQ(2d,q), DW(2d-1,q) and DH(2d-1,q 2 ), for d3; (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), n3.

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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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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