Blocking Sets in Desarguesian Projective Planes

Abstract

Using theorems of Redei, and of Brouwer and Schrijver, and Jamison, it is proved that a non-trivial blocking set in a desarguesian projective plane of order q has at least q + v (2q) + I points, if q is at least 7. odd and not a square and q i' 27. Further one can show that non-trivial blocking sets in the desarguesian planes PG(2, 11) and PG(2, 13) have at least 18 resp. 21 points, and this is best possible. In addition a nice description of a blocking set of size qt +qt-1 +I in the desarguesian plane PG(2, q') is given, where q is some prime power.

Blocking sets in PG(2, qn) from cones of PG(2n, q)

“…A. Bruen [8] was proved also for non Desarguesian finite projective planes. Moreover, in the case n > 2, improved results have been obtained by L.Storme and Sz.Weiner [26].…”

“…For example, in [6] a blocking set in PG(n, q) is defined as a 1−blocking set. For information on main results and recent developments of blocking set theory we refer the reader to [5,8,17,22,23,27,29,30]. Here we will survey just some results useful in what follows.…”

“…[8] for n = 2; A. Beutelspacher [6] for n > 2) The minimum possible size of a blocking set B in a finite projective space P G(n, q), n ≥ 2, is q + √ q + 1 and the bound is attained if, and only if, q is a square and B is a Baer subplane.…”

Explicit Constructions

“…225-226), which asserts that the number of directions m determined by a set of q points, not all on a line, in the affine plane AG(2, q), where q = p n is a prime power, is at least Blokhuis and Brouwer (1986) found a nice way to combine this result with the JamisonBrouwer-Schrijver Theorem (see Theorem 6.5), and derive a bound for the size of non-trivial blocking sets in desarguesian projective planes. Let P G(2, q) denote the projective plane over GF (q).…”

On minimum size blocking sets of external lines to a quadric in PG(d,q)

“…Blocking sets have been investigated by a great variety of authors, from many points of view [5,8,9]. Now, let G be a set of lines of P. A point set B of P is a blocking set with respect to G (or a G-blocking set) if every line in G is incident with at least one point of B.…”