Maximal partial spreads of the sizes 13, 14, 15,..., 22 and 26 are described. They were found by using a computer. The computer also made a complete search for maximal partial spreads of size less then or equal to 12. No such maximal partial spreads were found.

We prove the non-existence of maximal partial spreads of size 76 in PG(3,9). Relying on the classification of the minimal blocking sets of size 15 in PG(2, 9) [22], we show that there are only two possibilities for the set of holes of such a maximal partial spread. The weight argument of Blokhuis and Metsch [3] then shows that these sets cannot be the set of holes of a maximal partial spread of size 76. In [17], the non-existence of maximal partial spreads of size 75 in PG(3,9) is proven. This altogether proves that the largest maximal partial spreads, different from a spread, in PG(3, q = 9) have size q(2) - q + 2 = 74.

4.

Heden, Olof

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

An (s, t)-spread in a finite vector space V = V (n, q) is a collection F of t-dimensional subspaces of V with the property that every s-dimensional subspace of V is contained in exactly one member of F. It is remarkable that no (s, t)-spreads has been found yet, except in the case s = 1. In this note, the concept a-point to a (2,3)-spread F in V = V(7, 2) is introduced. A classical result of Thomas, applied to the vector space V, states that all points of V cannot be alpha-points to a given (2, 3)-spread.F. in V. In this note, we strengthened this result by proving that every 6-dimensional subspace of V must contain at least one point that is not an a-point to a given (2, 3)-spread of V.