Hans-Joachim Kroll

PD-sets for the codes related to some classical varieties

We generalize the notion of a PD-set of a code to that of a t-PD-set of an arbitrary permutation set. We find PD-sets for miquelian Benz planes of small order and for the ruled rational normal surface of order 3 in PG(4,3) and in PG(4,4). These results yield PD-sets for the related linear codes.

Perspectivities in Circle Geometries

During recent years research in the theory of circle geometries has made enormous progress. This development is most impressively reflected by the bibliography of BENZ’s book [4]. Circle geometries, as the term is used in our investigations, were first considered in the last century in the works of MOBIUS, LIE and LAGUERRE. It was not until the paper of B. L. van der WAERDEN and L. J. SMID that modern methods were introduced in the research of circle geometries. Also PETKANTSCHIN investigated circle geometries in 1940. Both papers received no attention for quite a while due to difficult times. In the fifties R. FURCH suggested the study of circle geometries and two of his advanced students (EWALD, BENZ) took up this suggestion to do work for a doctorate. Especially the beautiful results obtained by BENZ stimulated research in this field so that an increasing number of mathematicians started work in the field of circle geometries. Among those who made important contributions we would like to mention R. ARTZY, P. DEMBOWSKI, W. HEISE, H. MAURER, P. QUATTROCCHI.

Pseudo-Euklidische Ebenen und euklidische Räume

Euclidean planes and spaces as well as pseudo-euclidean planes are characterized solely by their incidence structure and a congruence relation on the set of the pairs of points.

Antiblocking systems and PD-sets

Since the permutation decoding algorithm is more efficient the smaller the size of the PD-set, it is important for the applications to find small PD-sets. A lower bound on the size of a PD-set is given by Gordon. There are examples for PD-sets, but up to now there is no method known to find PD-sets. The question arises whether the Gordon bound is sharp. To handle this problem we introduce the notion of antiblocking system and we show that there are examples where the Gordon bound is not sharp.

Zur Darstellung der Unterräume von Kettengeometrien

A subspace of a chain geometry in the sense of W. Benz [1] is defined as a subset S of points such that the chain joining any three pairwise non-parallel points of S is contained in S. In this note we are concerned with the problem of determining those subspaces that can be represented by subalgebras.

Perspektivitäten in Benz-Ebenen

In jeder Benz-Ebene (P, K) (d.i. eine Mobius-, Laguerre- oder Minkowski-Ebene [1])1 lassen sich zwei Arten von Perspektivitaten definieren (vgl. [4], [11]), einerseits die Beruhrtransformationen (vgl. § 2) und andererseits die Perspektivitaten mit einem Zentrum (vgl. [11]). Es bezeichne V bzw. P die Menge der Beruhrtransformationen bzw. der Perspektivitaten mit einem Zentrum. Jede Abbildung, die sich als Kette von Abbildungen aus V ⋃ P bzw. P bzw. V schreiben last, heist Projektivitat bzw. eigentliche Projektivitat bzw. Beruhrprojektivitat. Mit Γ(V ⋃ P) bzw. Γ(P) bzw. Γ(V) bezeichnen wir die Gruppe der Projektivitaten bzw. eigentlichen Projektivitaten bzw. Beruhrprojektivitaten eines Kreises K auf sich.

Eine Kennzeichnung der miquelschen Minkowski-Ebenen durch Transitivitätseigenschaften

A characterization of miquelian Minkowski planes by means of transitivity properties is given. As a corollary we obtain that in the classification of M. Klein, class 22 is empty.

Zur Darstellung Miquelscher Möbiusräume

In this paper we prove the following theorem (shown by J.TITS [7] in the case of finite dimension): LetO be a set of at least 2 points of a projective space π of dimension ≥ 2 and suppose that the intersections ofO and the planes of π are either empty or one point sets or Pascalian ovals. ThenO is an ellipsoid (nondegenerate quadric of index 1). As a corollary we obtain that every Miquelian Möbius-space is the geometry of the plane sections of an ellipsoid.

Ordnungsfunktionen in Miquelschen Möbiusebenen und Ihre Beziehungen zu Algebraischen Anordnungen

In this paper we investigate the relations between the geometrical ordering of a Miquelian Möbius-planeM(K, Q) and the ordering of the underlying field K. Every semi-ordering of K induces an order function ofM(K, Q), and conversely, if ¦k¦ >2, every semi-ordering of K is induced by exactly two (∞)-normal order functions ofM(K, Q). Moreover the normal order functions correspond to. those semi-orderings > of K for which is Q(x, y) >0 for all (x, y) ε (K × K){(0, 0)}. Full orderings ofM(K, Q) are defined similar to those in projective geometry.

Note: Construction of caps by means of caps in complementary subspaces

A construction of caps is given which yields in particular caps with a free pair of points. Applying this construction, we meet the bound of Farr and Lisonek [J. Farr, P. Lisonek, Caps with free pairs of points, J. Geom. 85 (2006) 35-41] for caps with a free pair of points in PG(5,q), q even.