Proofs of two conjectures on the dimensions of binary codes

Let ${\mathcal{L}}$ and ${\mathcal{L}_0}$ be the binary codes generated by the column ${\mathbb{F}_2}$ -null spaces of the incidence matrices of external points versus passant lines and internal points versus secant lines with respect to a conic in PG(2, q), respectively. We confirm the conjectures on the dimensions of ${\mathcal{L}}$ and ${\mathcal{L}_0}$ using methods from both finite geometry and modular representation theory.