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

Source 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 ( 2 d , q ) , DW ( 2 d - 1 , q ) and DH ( 2 d - 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 - ( 2 n - 1 , q ) , n ⩾ 3 .