Incidence Matrices of Polarized Projective Spaces

TL;DR

This paper studies the incidence matrices of polarized projective spaces over finite fields, classifying lines via quadratic forms and analyzing their ranks.

Contribution

It introduces a new classification of lines in Fq^4 using quadratic forms and proves the 2-rank of associated sub-matrices.

Findings

Defined a non-degenerate symmetric bilinear form on Fq^4

Classified lines as isotropic or anisotropic using quadratic form Q

Proved the 2-rank of the sub-matrices derived from the incidence matrix

Abstract

In this paper, we first define a non-degenerate symmetric bilinear form on Fq^4. Then we get an incidence matrix G of Fq^4 by the bilinear form. By its corresponding quadratic form Q, the lines of Fq^4 are classified as isotropic and anisotropic lines. Under this classification, we can get two sub-matrices of G and prove their 2-rank. Key words and phrases: finite field, quadratic form, incidence matrix.