New families of flag-transitive linear spaces (Q2689361)

Flag-transitive finite linear spaces are classified, up to the case of an automorphism group acting as a one-dimensional affine group in the O'Nan-Scott classification of primitive groups. The authors of the paper under review show once again that this case will be really hard by constructing new examples. Their method is based on a construction by \textit{M. Pauley} and \textit{J. Bamberg} [Finite Fields Appl. 14, No. 2, 537--548 (2008; Zbl 1137.51007)], using certain irreducible polynomials. In the paper under review, the authors use permutation polynomials of the form \(x^rh(x^{(q-1)/s})\) for appropriate \(r,s\). They also construct an extra example in characteristic 3 using a certain irreducible polynomial of degree 3.