Cameron-Liebler line classes

TL;DR

This paper introduces new Cameron-Liebler line classes in projective 3-space for many odd q, constructed via computer search, expanding the known examples and exploring their connections to geometric and algebraic structures.

Contribution

The paper provides new examples of Cameron-Liebler line classes with parameter (q^2 - 1)/2 for many odd q, constructed through computational methods and group actions.

Findings

New Cameron-Liebler line classes for many odd q

Construction via union of line orbits under cyclic collineation groups

Connections to two-intersection sets in affine planes

Abstract

New examples of Cameron-Liebler line classes in $\mathrm{PG}(3,q)$ are given with parameter $\frac{1}{2}(q^2 -1)$. These examples have been constructed for many odd values of $q$ using a computer search, by forming a union of line orbits from a cyclic collineation group acting on the space. While there are many equivalent characterizations of these objects, perhaps the most significant is that a set of lines $\mathcal{L}$ in $\mathrm{PG}(3,q)$ is a Cameron-Liebler line class with parameter $x$ if and only if every spread $\mathcal{S}$ of the space shares precisely $x$ lines with $\mathcal{L}$. These objects are related to generalizations of symmetric tactical decompositions of $\mathrm{PG}(3,q)$, as well as to subgroups of $\mathrm{P\Gamma L}(4,q)$ having equally many orbits on points and lines of $\mathrm{PG}(3,q)$. Furthermore, in some cases the line classes we construct are related to two-intersection sets in $\mathrm{AG}(2,q)$. Since there are very few known examples of these sets for $q$ odd, any new results in this direction are of particular interest.