New families of flag-transitive linear spaces

TL;DR

This paper introduces new families of flag-transitive linear spaces with specific point and line configurations, constructed via permutation polynomials and affine automorphisms, expanding the understanding of symmetric combinatorial structures.

Contribution

It presents novel constructions of flag-transitive linear spaces using permutation polynomials and affine automorphisms, extending previous classification schemes.

Findings

New families of linear spaces with specific parameters

Connection established between permutation polynomials and automorphism groups

Extension of existing construction methods for flag-transitive spaces

Abstract

In this paper, we construct new families of flag-transitive linear spaces with $q^{2n}$ points and $q^{2}$ points on each line that admit a one-dimensional affine automorphism group. We achieve this by building a natural connection with permutation polynomials of $\mathbb{F}_{q^{2}}$ of a particular form and following the scheme of Pauley and Bamberg in [A construction of one-dimensional affine flag-transitive linear spaces, Finite Fields Appl. 14 (2008) 537-548].