Mathematics – Combinatorics
Scientific paper
2009-11-10
Mathematics
Combinatorics
36 pages
Scientific paper
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 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 \cite{keith1} to produce several classes of structured low-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 matrix between the secant lines and the external points to $\mathcal{O}$. In this paper, we prove the conjecture on the dimension of $\mathcal{L}$ by using a combination of techniques from finite geometry and modular representation theory.
Sin Peter
Wu Junhua
Xiang Qing
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