Proofs of Two Conjectures On the Dimensions of Binary Codes

TL;DR

This paper proves two conjectures regarding the dimensions of specific binary codes derived from incidence structures in finite projective planes, using finite geometry and modular representation theory.

Contribution

It provides the first rigorous proof of conjectures on the dimensions of codes associated with conic incidences in projective planes, combining geometric and algebraic methods.

Findings

Confirmed the conjectures on code dimensions

Established connections between geometry and modular representation theory

Enhanced understanding of binary codes from finite geometries

Abstract

Let $\mathcal{L}$ and $\mathcal{L}_0$ be the binary codes generated by the column $\mathbb{F}_2$-null space of the incidence matrix of external points versus passant lines and internal points versus secant lines with respect to a conic in $PG(2, q)$, respectively. We confirm the conjectures on the dimensions of $\mathcal{L}$ and $\mathcal{L}_0$ using methods from both finite geometry and modular representation theory.