  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].
