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Blocking sets in \(PG(2,q)\) for small \(p\), and partial spreads in \(PG(3,7)\). (English) Zbl 1044.51006

A nontrivial blocking set in a projective plane is a set of points not containing any line and meeting each line of the plane. A. Blokhuis [Combinatorica 14, 111–114 (1994; Zbl 0803.05011)] proved that a nontrivial blocking set of \(PG(2,p)\), \(p\) prime, consists of at least \(3(p+1)/2\) points and such blocking sets actually exist for all values of \(p\). A blocking set \({\mathcal S}\) in \(PG(2,p ^ h)\) is called of Rédei type if there exists a line \(L\) such that \(| {\mathcal S} \setminus L | = p ^ h\); a non Rédei blocking set is called sporadic if it consists exactly of \(3(p+1)/2\) points. A unique sporadic blocking set was known and it lies in \(PG(2,7)\).
In the paper under review the authors find another example in \(PG(2,13)\) and establish using a personal computer that there are no other examples for \(PG(2,p)\), with \(p\) prime and \(p \leq 41\). Again with the help of a personal computer, the authors investigate certain partial spreads (i.e. sets of skew lines) of \(PG(3,7)\), since a nontrivial blocking set of \(PG(2,q)\) can be associated to any maximal partial spread of \(PG(3,q)\).

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs
51A40 Translation planes and spreads in linear incidence geometry

Citations:

Zbl 0803.05011
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