Designs and their codes
LDPC codes generated by conics in the classical projective plane
Designs, Codes and Cryptography
On binary codes from conics in PG(2,q)
European Journal of Combinatorics
Proofs of two conjectures on the dimensions of binary codes
Designs, Codes and Cryptography
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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.