Translation planes of odd order via Dembowski--Ostrom polynomials

Abstract

We describe a class of translation planes whose orders are of the form $q^{n}$, where $n$ is odd and $q$ is an odd prime power $>3$. These planes have the property that a translation complement fixes a triangle and acts transitively on the set of non-vertices of each side. The planes form an odd order analogue to the planes of Kantor--Williams [17] which have even order. The construction of the planes is based on a certain type of Dembowski--Ostrom polynomials.