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

• Published in: Finite fields and their applications 2022-09, Vol.82, p.102047, Article 102047
• Main Authors: De Beule, J., Mannaert, J.
• Format: Article
• Language: English
• Subjects: Affine space; Cameron-Liebler line class; Modular equality; Projective space
• ISSN: 1071-5797; 1090-2465
• DOI: 10.1016/j.ffa.2022.102047

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.