A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension

TL;DR

This paper establishes a modular equality condition for Cameron-Liebler line classes in projective and affine spaces of odd dimension, extending previous results and providing new constraints on their parameters.

Contribution

It generalizes a known modular equality for Cameron-Liebler line classes from three dimensions to higher-dimensional projective and affine spaces.

Findings

The modular equality excludes about half of the possible parameters.

The modular equality in affine spaces is a stronger condition than in projective spaces.

The results provide new restrictions on the existence of Cameron-Liebler line classes.

Abstract

In this article we study Cameron-Liebler line classes in PG$(n,q)$ and AG$(n,q)$, objects also known as boolean degree one functions. A Cameron-Liebler line class $\mathcal{L}$ is known to have a parameter $x$ that depends on the size of $\mathcal{L}$. One of the main questions on Cameron-Liebler line classes is the (non)-existence of these sets for certain parameters $x$. In particularly it is proven in [12] for $n=3$, that the parameter $x$ should satisfy a modular equality. This equality excludes about half of the possible parameters. We generalize this result to a modular equality for Cameron-Liebler line classes in PG$(n,q)$, and AG$(n,q)$ respectively. Since it is known that a Cameron-Liebler line class in AG$(n,q)$ is also a Cameron-Liebler line class in its projective closure, we end this paper with proving that the modular equality in AG$(n,q)$ is a stronger condition than the condition for the projective case.