Improved upper bounds for partial spreads

A partial ((k-1))-spread in ({text {PG}}(n-1,q)) is a collection of ((k-1))-dimensional subspaces with trivial intersection. So far, the maximum size of a partial ((k-1))-spread in ({text {PG}}(n-1,q)) was known for the cases (nequiv 0pmod k), (nequiv 1pmod k), and (nequiv 2pmod k) with the additional requirements (q=2) and (k=3). We completely resolve the case (nequiv 2pmod k) for the binary case (q=2).