  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 is
provided. The construction uses both geometry and finite fields theory and
generalizes to any $q$, yielding a new family of $q$-ary
$(6,q^6+2q^2+2q+1,4;3)$ subspace codes.
