Subspaces intersecting in at most a point (Q2302159)

The author finds a new lower bound on the maximum number of planes in \(\mathrm{PG}(8,q)\) pairwise intersecting in at most a single point. This result has direct consequences for constant dimension codes, namely it follows that \[A_q(9,4;3) \geq q^{12} + 2q^8 + 2q^7 + q^6 + 2q^5 + 2q^4 -2q^2 -2q +1.\]