Dimensions of some binary codes arising from a conic in

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 Droms et al. (2006) [6] to produce several classes of structured low-density parity-check binary codes. In particular, the authors of Droms et al. (2006) [6] gave conjectured dimension formula for the binary code L which arises as the F 2 -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.