On Saturating Sets in Small Projective Geometries

A set of points, S ⊆ PG(r, q), is said to be %-saturating if, for any point x ∈ PG(r, q), there exist %+ 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, %). 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) ≤ 3q + 1 for q = 2, q ≥ 4. We further give an upper bound on k(% + 1, pm , %).