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
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,q6+2q2+2q+1,4;3) subspace codes