2021/07/12 by John Bamberg, Bamberg, John, Jesse Lansdown +1 · 1 citation
Computer Science · Engineering · Mathematics · #05B25 #05B30 #05E30 #51E05 #51E12 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2107.05207
openalex publication_date 2021/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \not ∈ T and all i, j \not ∈ 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\geqslant 3.