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

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 L is known to have a parameter x that depends on the size of L. One of the main questions on Cameron-Liebler line classes is the (non)-existence of these sets for certain parameters x. In particular it is proven in [14] 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), n≥7 odd, respectively AG(n,q), n≥3 odd. 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.