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

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 \(q^2- q + 1 – \bar{t}\bar{z}\), for every pair \((\bar{t},\bar{z}) \in P_1 \cup P_2\), where \(P_1\) and \(P_2\) are the pair sets \(P_1=\{(t,z)\in\mathbb{Z}\times\mathbb{Z}:\frac{q}{2}-2\leq t\leq q-3,0\leq z\leq\frac{q}{2}-2\}\) and \(P_2=\{(t,z)\in\mathbb{Z}\times\mathbb{Z}:0\leq t\leq \frac{q}{2}-3,0\leq z\leq{q}-1\}\), for \(q \geq 8\).