{"title":"Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry","authors":"John Bamberg, Jesse Lansdown","doi":"10.5802/alco.246","DOIUrl":null,"url":null,"abstract":"In this paper we show that if θ is a T-design of an association scheme (Ω,ℛ), and the Krein parameters q i,j h 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,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 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.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":"39 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebraic Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/alco.246","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper we show that if θ is a T-design of an association scheme (Ω,ℛ), and the Krein parameters q i,j h 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,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 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.