About maximal partial 2-spreads in $\mathrm{PG}(3m-1,q)$

Abstract

In this article we construct maximal partial 2-spreads in $PG\left(8,q\right)$ of deficiency $\delta =\left(k-1\right)\cdot q^2$, where $k\le q^2+q+1$ and $\delta =k\cdot q^2+l\cdot \left(q^2-1\right)+1$, where $k+l\le q^2$ and $\delta =\left(k+1\right)\cdot q^2+l\cdot \left(q^2-1\right)+m\cdot \left(q^2-2\right)+1$, where $k+l+m\le q^2$. Using these results, we also construct maximal partial 2-spreads in $PG\left(3m-1,q\right)$ of various deficiencies for $m\ge 4$.