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

The partial plane spreads in $${{\,\mathrm{PG}\,}}(6,2)$$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 $${{\,\mathrm{PG}\,}}(6,2)$$PG(6,2) of type $$(3^{16} 4^1)$$(31641), binary $$3\times 4$$3×4 MRD codes of minimum rank distance 3, and subspace codes with the optimal parameters $$(7,17,6)_2$$(7,17,6)2 and $$(7,34,5)_2$$(7,34,5)2.

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