On Binary Codes from Conics in PG(2,q)

TL;DR

This paper investigates the algebraic and geometric properties of a binary code derived from conics in projective plane PG(2,q), establishing its dimension through finite geometry and modular representation theory.

Contribution

It introduces a novel approach combining finite geometry and modular theory to determine the dimension of the binary code from conics, confirming a conjecture.

Findings

Dimension of the null space L computed

Proof of the conjecture on code dimension

Enhanced understanding of codes from conics

Abstract

Let A be the incidence matrix of passant lines and internal points with respect to a conic in PG(2, q), where q is an odd prime power. In this article, we study both geometric and algebraic properties of the column null space L of A over the finite field of 2 elements. In particular, using methods from both finite geometry and modular presentation theory, we manage to compute the dimension of L, which provides a proof for the conjecture on the dimension of the binary code generated by L.