A family of translation planes of order \(q^{2m+1}\) with two orbits of length 2 and \(q^{2m+1}-1\) on \(l_ \infty\)
The author constructs a class of translation planes of order \(q^{2m+1}\) where \(q\) is an odd prime power and \(m\geq 1\). These translation planes \(\pi\) have the following properties: \(\pi\) has order \(2m+1\) over its kernel. If \(q^{2m+1}\neq 3^ 3\) then the linear translation complement has two orbits on \(\ell_ \infty\): one of length 2 and the other of length \(q^{2m+1}-1\). Many of these planes are not generalized André planes. If \(m=1\) then the above class of translation planes is contained in the class of planes constructed by Hiramine in 1986. If \(q^{2m+1}=3^ 3\) then each plane of this class is isomorphic to the Hering plane of order 27.
scientific article; zbMATH DE number 57581