A note on Cameron - Liebler line classes in PG(n,4)

TL;DR

This paper classifies Cameron-Liebler line classes in PG(3,4) and extends the classification to higher dimensions PG(n,4) for n ≥ 4, advancing understanding of these geometric structures.

Contribution

It provides a complete classification of Cameron-Liebler line classes in PG(3,4) and generalizes the results to PG(n,4) for all n ≥ 4, which was previously unknown.

Findings

Complete classification in PG(3,4)

Extension of classification to PG(n,4), n ≥ 4

New insights into the structure of Cameron-Liebler line classes

Abstract

A {\it Cameron -- Liebler line class} ${\cal L}$ with parameter $x$ is a set of lines of projective geometry $PG(3,q)$ such that each line of ${\cal L}$ meets exactly $x(q+1)+q^2-1$ lines of ${\cal L}$ and each line that is not from ${\cal L}$ meets exactly $x(q+1)$ lines of ${\cal L}$. In this paper, we obtain a classification of Cameron -- Liebler line classes in PG(3,4) and a classification of their generalization in PG(n,4), $n\geqslant 4$.