  Let A be the incidence matrix of passant lines and internal points with
respect to a conic in PG(2, q), where q is an odd prime power. In this article,
we study both geometric and algebraic properties of the column null space L of
A over the finite field of 2 elements. In particular, using methods from both
finite geometry and modular presentation theory, we manage to compute the
dimension of L, which provides a proof for the conjecture on the dimension of
the binary code generated by L.
