New upper bounds on the smallest size of a saturating set in a projective plane

Abstract

In a projective plane Π q (not necessarily Desar-guesian) of order q, a point subset S is saturating (or dense) if any point of Π q \S is collinear with two points in S. Using probabilistic methods, more general than those previously used for saturating sets, the following upper bound on the smallest size s(2, q) of a saturating set in Π q is proved: s(2, q) 1 a random point set of size k in Π q with 2c√(q + 1) ln(q + 1) + 2 q is (1,μ)-saturating if for every point Q of Π q \S the number of secants of S through Q is at least μ, counted with multiplicity. The multiplicity of a secant l is computed as (#(l∩S) 2 ). The following upper bound on the smallest size s μ (2, q) of a (1,μ)-saturating set in Π q is proved: s μ (2, q) < 2(μ + 1)√(q + 1)ln(q + 1) + 2 for 2 < μ < √q. By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a (1, μ)-saturating set) in the projective space PG(N, q) are obtained. All the results are also stated in terms of linear covering codes.