New families of flag-transitive linear 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].