De Beule, Jan

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Ghent University Academic Bibliography

The smallest minimal blocking sets of Q(2n, q), forsmall odd qJan De BeuleGhent UniversityJoint work with: Leo StormeIn [2] we used results on the size of the smallest minimal blocking sets ofQ(4, q), q even (from [1]) and projection arguments the find the following char-acterization of the smallest minimal blocking sets of Q(6, q), q even, q > 32:Theorem 1 Let K be a minimal blocking set of Q(6, q), different from an ovoidof Q(6, q), |K| 6 q3 + q. Then there is a point p ∈ Q(6, q) \K with the followingproperty: Tp(Q(6, q)) ∩ Q(6, q) = pQ(4, q)and K consists of all the points ofthe lines L on p meeting Q(4, q) in an ovoid O, minus the point p itself, and|K| = q3 + q.The results of [1] for Q(4, q), q even, could be extended to q = 3, 5, 7. Thenusing the same projection arguments we proved the above characterization forq = 3, 5, 7.Using inductive arguments we can find analogous results for Q(2n, q), q =3, 5, 7. The situation is now dependent on q, since Q(6, 3) has an ovoid, butQ(6, q), q = 5, 7, not. For q = 5, 7, we proved the following characterization.Theorem 2 Let K be a minimal blocking set of Q(2n + 2, q), n > 2, |K| 6qn+1 + qn−1. Then there is an (n − 2)-dimensional space pi, pi ⊂ Q(2n + 2, q),pi∩K = ∅, with the following property: Tpi(Q(2n+2, q))∩Q(2n+2, q) = piQ(4, q)and K is a cone with vertex pi and base O, where O is an ovoid of Q(4, q), minusthe points of the vertex pi, and |K| = qn+1 + qn−1.For q = 3 we proved a characterization using ovoids of Q(6, 3).Theorem 3 Let K be a minimal blocking set of Q(2n + 2, 3), n > 3, |K| 6qn+1 + qn−2. Then there is an (n − 3)-dimensional space pi, pi ⊂ Q(2n + 2, 3),pi∩K = ∅, with the following property: Tpi(Q(2n+2, 3))∩Q(2n+2, 3) = piQ(6, 3)and K is a cone with vertex pi and base O, where O is an ovoid of Q(6, 3), minusthe points of the vertex pi, and |K| = qn+1 + qn−2.We will discuss several aspects of the theorems and the difficulties which arisefor other values of q.References1. J. Eisfeld, L. Storme, T. Szo˝nyi, and P. Sziklai, Covers and blocking setsof classical generalized quadrangles, Discrete Math., 238(1-3):35–51, 2001.2. J. De Beule and L. Storme, The smallest minimal blocking sets of Q(6, q),q even, J. Combin. Des., to appear.

The smallest minimal blocking sets of Q(2n+2,q), for small odd q

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The smallest minimal blocking sets of Q(2n+2,q), for small odd q

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Ghent University Academic Bibliography