Dimensions of some binary codes arising from a conic in PG(2, q)

Let O be a conic in the classical projective plane PG(2, q), where q is an odd prime power. With respect to O, the lines of PG(2, q) are classified as passant, tangent, and secant lines, and the points of PG(2, q) are classified as internal, absolute and external points. The incidence matrices between the secant/passant lines and the external/internal points were used in [5] to produce several classes of structured low-density parity-check binary codes. In particular, the authors of [5] gave conjectured dimension formula for the binary code L which arises as the F2-null space of the incidence matrix between the secant lines and the external points to O. In this paper, we prove the conjecture on the dimension of L by using a combination of techniques from finite geometry and modular representation theory.