Ovoids and Caps in Planar Spaces

Abstract

It is supposed that a family p of subspaces in (S, L) exists such that |p| ≥ 2, every π ∈ p contains three independent points, and through three independent points, there is only one element of p. The triple (S, L, p) is called “planar space,” the elements of p are called “planes.” Examples of planar spaces are (1) the affine or projective spaces of dimension n ≥ 3, with respect to their lines and their planes and (2) any subset H in PG(n, q), n ≥ 3, [H] = PG(n, q), with respect to the intersections of H with its secant lines and planes having three independent points of H. This chapter provides an example by considering a family p′ of d-dimensional subspaces in PG(n, q), n ≥ 4, two by two meeting in at most one line, and p′′ is considered the family of planes in PG(n, q), each of them not belonging to any S d ∈ p′. It is supposed that the lines of the planar space (S, L, p) have the same size k ≥ 3, and the planes have the same size ν. Then (S, L) is a Steiner system S(2, k, V), |S| = V, and for any π ∈ p, if L π denotes the set of lines in π, (π, L π ) is a Steiner system S (2, k, v).