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

TL;DR

This paper proves a conjecture regarding the dimension of a specific binary code derived from the incidence structure of points and lines related to a conic in projective plane geometry, combining finite geometry and representation theory.

Contribution

It establishes the conjectured dimension formula for a binary code associated with a conic in $PG(2,q)$, using novel techniques from finite geometry and modular representation theory.

Findings

Confirmed the conjectured dimension formula for the code

Connected finite geometry with modular representation theory

Enhanced understanding of incidence matrices in projective planes

Abstract

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.