New Examples of Maximal Partial Line Spreads in $PG(4,q)$

Abstract

We construct a class of maximal partial line spreads in $\mathrm{P}\mathrm{G}(4,q)$, that we call $q$-added maximal partial line spreads. We obtain them by depriving a line spread of a hyperplane of some lines and adding $q+1$ pairwise skew lines not in the hyperplane for each removed line. We do it in a theoretical way for every value of $q$, and by a computer search for $q\le 16$. More precisely, we prove that for every $q$ there are $q$-added MPS of size $q^2+kq+1$, for every integer $k=1,\dots ,q-1$, while by a computer search we get larger cardinalities.