Subspace code constructions

We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\mathcal{A}}_q(9,4;3)\ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant dimension subspace codes. We also construct two new non-equivalent $(6,(q^3-1)(q^2+q+1),4;3{)}_q$ constant dimension subspace orbit-codes.