On Saturating Sets in Small Projective Geometries

A set of points, S ⊆ PG(r, q), is said to be ϱ -saturating if, for any point x ∈ PG(r, q), there exist ϱ + 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q + 1 forq = 2, q ≥ 4. We further give an upper bound onk (ϱ + 1, pm, ϱ).