Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$

TL;DR

This paper introduces a new algebraic construction of Cameron-Liebler line classes with specific parameters for certain prime powers, and also constructs affine two-intersection sets in affine planes for particular cases.

Contribution

It provides the first algebraic construction of an infinite family of Cameron-Liebler line classes with parameter x=(q^2-1)/2 for q ≡ 5 or 9 mod 12, and constructs affine two-intersection sets when q is a power of 3.

Findings

New infinite family of Cameron-Liebler line classes for specified q.

First construction of affine two-intersection sets in AG(2,q) for q a power of 3.

Generalizes previous computational examples by Rodgers.

Abstract

In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$ for $q\equiv 5$ or $9\pmod{12}$, which generalizes the examples found by Rodgers in \cite{rodgers} through a computer search. Furthermore, in the case where $q$ is an even power of $3$, we construct the first infinite family of affine two-intersection sets in $\mathrm{AG}(2,q)$.