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

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) <; 2√(q + 1)ln(q + 1) + 2 ~ 2√q ln q. We also show that for any constant c > 1 a random point set of size k in Π
_q
with 2c√(q + 1) ln(q + 1) + 2 <; k <; q
²
-1/q+2 ~ q is a saturating set with probability greater than 1 - 1/(q + 1)
^{2c2 -2}
. Our probabilistic approach is also applied to multiple saturating sets. A point set S ⊂ Π
_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.