On the non-existence of a maximal partial spread of size 76 in PG(3, 9)

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.