Classification of large partial plane spreads in $PG(6,2)$ and related combinatorial objects

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16}{4}^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6{)}_2$ and $(7,34,5{)}_2$.