On binary codes from conics in PG(2,q)

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 qq is an odd prime power. In this article, we study both geometric and algebraic properties of the column F2F2-null space LL of A. In particular, using methods from both finite geometry and modular presentation theory, we manage to compute the dimension of LL, which provides a proof for the conjecture on the dimension of the binary code generated by LL.

► We study algebraic and geometric properties of conics in PG(2,q). ► We confirm the conjecture on the dimension of the F2F2-null space of the incidence matrix of passant lines and internal points. ► The tools involved are some results on modular representations of PSL(2,q).