A new class of maximal partial line spreads in $PG(3,q),q$ even

Abstract

In this work we construct many new examples of maximal partial line spreads in $PG(3,q),q$ even. We do this by giving a suitable representation of $PG(3,q)$ in the non-singular quadric $Q(4,q)$ of $PG(4,q)$. We prove the existence of maximal partial line spreads of sizes $q^2-q+1–\overline{t}\overline{z}$, for every pair $(\overline{t},\overline{z})\in P_1\cup P_2$, where $P_1$ and $P_2$ are the pair sets $P_1=\{(t,z)\in \mathbb{Z}\times \mathbb{Z}:\frac{q}{2}-2\le t\le q-3,0\le z\le \frac{q}{2}-2\}$ and $P_2=\{(t,z)\in \mathbb{Z}\times \mathbb{Z}:0\le t\le \frac{q}{2}-3,0\le z\le q-1\}$, for $q\ge 8$.