  In this work we construct a new class of maximal partial spreads in
$PG(4,q)$, that we call $q$-added maximal partial spreads. We obtain them by
depriving a spread of a hyperplane of some lines and adding $q+1$ lines not of
the hyperplane for each removed line. We do this in a theoretic way for every
value of $q$, and by a computer search for $q$ an odd prime and $q \leq 13$.
More precisely we prove that for every $q$ there are $q$-added maximal partial
spreads from the size $q^2+q+1$ to the size $q^2+(q-1)q+1$, while by a computer
search we get larger cardinalities.
