Incidence Matrices of Finite Quadratic Spaces

TL;DR

This paper investigates the 2-rank of incidence matrices in finite projective spaces, classifies points via quadratic forms, and proposes a conjecture supported by computational evidence for specific cases.

Contribution

It introduces a classification of points using quadratic forms and formulates a conjecture on the 2-rank of sub-matrices in PG(n,q), with proofs for n=1, 2.

Findings

Proved the 2-rank of the full incidence matrix for PG(n,q) with q odd.

Classified points into isotropic and anisotropic using quadratic forms.

Conjectured the 2-rank of sub-matrices, supported by computational evidence.

Abstract

In this paper, we first prove the 2-rank of full incidence matrix of PG(n,q) with $q$ an odd prime power. Then by the quadratic form defined on PG(n,q), the points of it are classified as isotropic and anisotropic points. We divide the full incidence matrix into four sub-matrices. Then by using the software package Magma, we give a general conjecture for the 2-rank of sub-matrices of the full incidence matrix in PG(n,q) and prove it in the case of n=1, 2.