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 space of the incidence matrix 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.