Cameron-Liebler line classes in PG(3,4)

Abstract

Cameron-Liebler line classes are sets of lines in PG(3,q) that contain a fixed number $x$ of lines of every spread. Cameron and Liebler classified them for $x\in\{0,1,2,q^2-1,q^2,q^2+1\}$ and conjectured that no others exist. This conjecture was disproved by Drudge and his counterexample was generalised to a counterexample for any odd $q$ by Bruen and Drudge. In this paper, we give the first counterexample for even $q$, a Cameron-Liebler line class with parameter $7$ in PG(3,4). We also prove the nonexistence of Cameron-Liebler line classes with parameters $4$ and $5$ in PG(3,4) and give some properties of a hypothetical Cameron-Liebler line class with parameter $6$ in PG(3,4).