Some new invariant pairs (t,3) for projective Hjelmslev planes

Abstract

Associated with every finite projective Hjelmslev plane is an invariant pair (t, r): t is the number of neighbours of a given point on a given line passing through it and r is the order of the underlying projective plane. The Drake-Lenz method [2], [3] of using auxiliary matrices for the constructions of projective Hjelmslev planes has become standard by now. This paper is intended to give some new constructions of projective Hjelmslev planes with invariant pairs (t, 3) by making use of the generalization and improvement of the Drake-Lenz theorem [3] obtained by the author in [6] and [7]. The results of this paper add 8 new values to the list ([5], example 3.7(ii)) of invariant pairs (t, 3) with t ≤ 1,000 for projective Hjelmslev planes.