Ovoids, spreads and m-systems of finite classical polar spaces
J. A. Thas · Cambridge University Press eBooks · 2001
A survey of the most important results on partial m -systems and m -systems of finite classical polar spaces will be given. Also, the paper contains several recent results on the topic. Finally, many applications of m -systems to strongly regular graphs, linear projective two-weight codes, maximal arcs, generalized quadrangles and semi-partial geometries are mentioned. Introduction Let P be a finite polar space of rank r ≥ 2. An ovoid O of P is a pointset of P, which has exactly one point in common with each generator of P , that is, with each maximal totally singular subspace of P . A spread S of P is a set of generators, which constitutes a partition of the pointset. It appears that ∣ O ∣ = ∣S∣ for any ovoid O and any spread S of any given polar space P ; this common number will be denoted by μ P . Ovoids and spreads have many connections with and applications to projective planes, circle geometries, generalized polygons, strongly regular graphs, partial geometries, semi-partial geometries, codes, designs. A partial msystem of P , with 0 ≤ m ≤ r − 1, is any set {π 1 , π 2 ,…,π k } of ( k ≠0) totally singular m -spaces of P such that no generator containing π i has a point in common with (π 1 ∪ π 2 ∪ … ∪ π k ) − π i ,π k , with i = 1, 2,…, k . For any partial m -system M of P the bound ∣ M ∣ ≤ μ P holds. If ∣ M ∣ = μ P , then the partial m -system M of P is called an m -system of P .