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It is shown that the maximum size of a binary subspace code of packet length$v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$;in Finite Geometry terms, the maximum number of planes in$\operatorname{PG}(5,2)$ mutually intersecting in at most a point is $77$.Optimal binary $(v,M,d;k)=(6,77,4;3)$ subspace codes are classified into $5$isomorphism types, and a computer-free construction of one isomorphism type isprovided. The construction uses both geometry and finite fields theory andgeneralizes to any $q$, yielding a new family of $q$-ary$(6,q^6+2q^2+2q+1,4;3)$ subspace codes.

Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4