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Let $\mathcal{O}$ be a conic in the classical projective plane $PG(2,q)$,where $q$ is an odd prime power. With respect to $\mathcal{O}$, the lines of$PG(2,q)$ are classified as passant, tangent, and secant lines, and the pointsof $PG(2,q)$ are classified as internal, absolute and external points. Theincidence matrices between the secant/passant lines and the external/internalpoints were used in \cite{keith1} to produce several classes of structuredlow-density parity-check binary codes. In particular, the authors of\cite{keith1} gave conjectured dimension formula for the binary code$\mathcal{L}$ which arises as the $\Ff_2$-null space of the incidence matrixbetween the secant lines and the external points to $\mathcal{O}$. In thispaper, we prove the conjecture on the dimension of $\mathcal{L}$ by using acombination of techniques from finite geometry and modular representationtheory.

Dimensions of Some Binary Codes Arising From A Conic in $PG(2,q)$