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

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Abstract

In this paper we show that if θ is a T-design of an association scheme (Ω, R), and the Krein parameters qi, jh vanish for some h ∉ T and all i, j ∉ T (i, j, h ≠ 0), then θ consists of precisely half of the vertices of (Ω, 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, s2) 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, s2); (iii) the dual polar spaces DQ(2d, q), DW(2d - 1, q) and DH(2d-1, q2), for d ≥ 3; (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), n ≥ 3.

Original languageEnglish
Pages (from-to)197-212
Number of pages16
JournalAlgebraic Combinatorics
Volume6
Issue number1
DOIs
Publication statusPublished - 2023

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