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

Abstract

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 (3 164 1) , binary 3 × 4 MRD codes of minimum rank distance 3, and subspace codes with the optimal parameters (7 , 17 , 6) 2 and (7 , 34 , 5) 2.