Abstract. We find all minimal blocking sets of size \ (p + 1) in PG(2, p) for p < 41. There is one new sporadic example, for p -13. We find all maximal partial spreads of size 45 in PG(3,7).
Minimal nontrivial blocking sets in PG(2, p)A blocking set in a projective plane is a set of points meeting all lines. It is called nontrivial when it does not contain a line. An m-secant of a set is a line meeting the set in precisely m points.Blokhuis [2] shows that in a Desarguesian projective plane PG(2,/?) of prime order /?, a nontrivial blocking set has size at least \(p+ 1), and, moreover, that in case of equality each point of the blocking set lies on precisely ^(p -1) tangents (1 -secants).Nontrivial blocking sets of size | (p + 1) exist for all p. Indeed, an example is given by the projective triangle: the set consisting of the points (0, 1,-s 2 ), (l,-j 2 ,0), (-s 2 ,0, 1) with s e F^.No nontrivial blocking set of size q + m in PG(2, q) can have a fc-secant for k > w, and in particular such a set of size | (p + 1) in PG(2, p) cannot have a fc-secant with k>^(p + 3). The triangle has three ^(p 4-3)-secants. Conversely, Loväsz and Schrijver [10] show that any nontrivial blocking set of size | (p + 1) with a | (p -h 3)-secant must be protectively equivalent to the triangle. (They put the given secant at infinity and show that the remaining p affine points can be taken to be the points A blocking set S in PG(2, q) is called ofRedei type when there is a line L such that \S\L\ = q. Thus, we know the blocking sets ofRedei type meeting the Blokhuis bound in PG(2, p], p prime. Let us call a nontrivial blocking set in PG(2,/?) that meets the Blokhuis bound sporadic if it is not of Redei type. A single sporadic blocking set (in PG(2, 7)) was known. Here we find a second sporadic blocking set (in PG(2, 13)) and show that no other sporadic blocking sets exist in PG(2,/?), p < 41.