Flokki planes and cubic polynomials

Abstract

Non-Desarguesian translation planes of order $q^2$ are constructed whenever $q=2^e\ge 16$ and $e$ isnot divisible by $3$. Each plane has kernel $\textnormal{GF}(q)$  andtranslation complement of order $q(q-1)^2 e$, with orbits of lengths$1,$ $q$ and $q^2-q$ on the translation line. The planes have elation groups oforder $q$ that producederivable nets, but are not flock planes, semifield planes,  orlifted planes.
The same algebraic tools are used to construct  non-Desarguesiantranslation planes of order$2^p$ for every prime $p>3$.