On binary codes from conics in

Let A be the q ( q − 1 ) 2 × q ( q − 1 ) 2 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 F 2 -null space L of A . 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 .