Improved upper bounds for partial spreads

A partial $(k-1)$-spread in $\mathrm{PG}(n-1,q)$ is a collection of $(k-1)$-dimensional subspaces with trivial intersection, i.e., each point is covered at most once. So far the maximum size of a partial $(k-1)$-spread in $\mathrm{PG}(n-1,q)$ was known for the cases $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}k)$, $n\equiv 1\pmod k$and$n\equiv 2\pmod k$withtheadditionalrequirements$q=2$and$k=3$\mathrm{.}Wecompletelyresolvethecase$n\equiv 2\pmod k$forthebinarycase$q=2$.