A new class of translation planes of order \(q^ 3\)
Using the Oyama's method [\textit{T. Oyama}, Osaka J. Math. 22, 35-54 (1985; Zbl 0561.51006)] the author is concerned with the problem of describing a new class of translation planes of order \(q^ 3\), where 2 is a non- square in GF(q). He shows that for \(q\neq 3\) the linear translation complement of any plane of this class is of order \(3(q-1)(q^ 3-1)\) and for \(q=3\) the plane is the Hering plane of order 27 [\textit{C. Hering}, Abh. Math. Semin. Univ. Hamb. 34, 203-208 (1970; Zbl 0195.502)], and the translation complement is isomorphic to SL(2,13). The paper concludes by showing that the planes differ from generalized André planes [\textit{J. André}, Math. Z. 60, 156-186 (1954; Zbl 0056.385)] and semifield planes.
scientific article; zbMATH DE number 3917106