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 $q^2–q+2=74$.