On m-ovoids of regular near polygons

Abstract

We generalise the work of Segre (1965), Cameron -Goethals -Seidel (1978), and by showing that nontrivial m-ovoids of the dual polar spaces DQ(2d, q), DW(2d − 1, q) and DH(2d − 1, q 2 ) (d 3) are hemisystems. We also provide a more general result that holds for regular near polygons.2010 Mathematics Subject Classification. 05B25, 51E12, 51E20.

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

“…However, it has no hemisystems. While the dual polar space DQ(6, 3) also has Krein parameter q 3 33 = 0, and does have hemisystems (see [1]). The following is a long-standing open problem.…”

“…Cameron, Goethals, and Seidel [7] proved that strongly regular graphs with either q 1 11 = 0 or q 2 22 = 0 have strongly regular subconstituents about every vertex. Moreover, they characterised such graphs, when connected, as being a pentagon, a Smith graph (cf.…”

A generalization of the Haemers–Mathon bound for near hexagons

“…Note that rank( C) = rank( C(k)) if k = k. Suppose now that k = k + 1, or equivalently, t Remark. In [9], the matrix C was written as the sum of two matrices, where one of them was the "block diagonal matrix" with "diagonal entries" equal to F (1) , F (2) , . .…”

“…It also seems that near polygons form the natural setting for studying certain problems on (substructures of) dual polar spaces and generalized polygons, see e.g. [1].…”

On regular systems of finite classical polar spaces

“…A regular system with respect to (k − 1)-spaces of P d,e having the same size as its complement (in M P d,e ) is said to be a hemisystem with respect to (k − 1)-spaces of P d,e (of course here |M P d,e | must be even). From [6,47,53], a regular system w.r.t. (d − 2)-spaces of P d,e , d ≥ 3 and e ≤ 1, or (d, e) = (2, 1/2) is a hemisystem.…”