  We improve on the lower bound of the maximum number of planes in
$\operatorname{PG}(8,q)\cong\F_q^{9}$ pairwise intersecting in at most a point.
In terms of constant dimension codes this leads to $A_q(9,4;3)\ge q^{12}+
2q^8+2q^7+q^6+2q^5+2q^4-2q^2-2q+1$. This result is obtained via a more general
construction strategy, which also yields other improvements.
