On translation planes of order \(q^ 3\) with an orbit of length \(q^ 3- 1\) on \(\ell _{\infty}\) (Q1086835)

The author considers translation planes of order \(q^ 3\) (with kernel GF(q)) satisfying the condition (*) A subgroup G of the translation complement has orbits of size 2 and \(q^ 3-1\) on the line at infinity. He constructs a new class of such planes containing members for all odd values of q. He also shows the following: If \({\mathfrak P}\) is any translation plane satisfying (*) and if \({\mathfrak P}\) is not an André plane, then either the translation complement of \({\mathfrak P}\) has order \(3^ i(q^ 3-1)(q-1)\) with \(i=0,1,2\), or is isomorphic to SL(2,13). He finally gives a characterization of the planes constructed.