Finite {0,s,2s}{0,s,2s}-affine linear spaces

In [The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, (1993).] Beutelspacher and Kersten studied finite {0,s,t}{0,s,t}-semiaffine linear spaces, with t≠2st≠2s. They characterized those with different point degrees, and in the constant point degree case they obtained a nice characterization of {0,1,t}{0,1,t}-semiaffine linear spaces. If t=2st=2s, only the cases s=1,2s=1,2 have been studied [W. Hauptmann, Endliche [0,2][0,2]-Ebenen., Geom. Ded. 9, (1980), 77–86; P. M. Lo Re, D. Olanda, {0,2,4}{0,2,4}-semiaffine planes, in: Combinatorics ’88, vol. 2 (Ravello, 1988), Research Lecture Notes Mathematics Mediterranean, Rende, 1991, pp. 195–210.] In this paper we study the case t=2st=2s, s⩾2s⩾2. We determine the point degrees and the line sizes of a {0,s,2s}{0,s,2s}-semiaffine linear space, and we also give some characterization results.