  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.
