On the Non-Existence of a Maximal Partial Spread of Size \(76\) in \(PG(3, 9)\)

Abstract

We prove the non-existence of maximal partial spreads of size \(76\) in \(\text{PG}(3,9)\). Relying on the classification of the minimal blocking sets of size 15 in \(\text{PG}(2,9)\) \([22]\), 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 \(\text{PG}(3,9)\) is proven. This altogether proves that the largest maximal partial spreads, different from a spread, in \(\text{PG}(3,q = 9)\) have size \(q^2 – q + 2 = 74\).