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Bruck nets with a transitive direction: to Helmut Salzmann on the occasion of his 60th birthday ()

We start the systematic investigation of the geometric properties and the collineation groups of Bruck nets N with a transitive direction (i.e. with a group G of central translations acting transitively on each line of a given parallel class P). After reviewing some basic properties of such nets (in particular, their connection to difference matrices), we shall consider the problem of what can be said if either N or G admits an interesting extension. Specifically, we shall handle the following four situations: (1) there is a second transitive direction; (2) N is a translation net (w.1.o.g. with translation group K containing G); (3) the dual of N \ P is a translation transversal design (w.1.o.g. with translation group K containing G); (4) N admits a transversal (and can then in fact be extended by adding a further parallel class). Our study of these problems will yield interesting generalizations of known concepts (e.g. that of a fixed-point-free group automorphism) and results (for affine and projective planes). We shall also see that a wide variety of seemingly unrelated results and constructions scattered in the literature are in fact closely related and should be viewed as part of unified whole.

Translation nets: a survey ()

In this survey, we will discuss the existence problem for translation nets, the question of when a translation net is maximal and the codes of abelian translation nets.

Characterizing Geometric Designs ()

We conjecture that the classical geometric 2-designs formed by the points and d-dimensional subspaces of the projective space of dimension n over the field with q elements, where 2 <= d <= n-1, are characterized among all designs with the same parameters as those having line size q+1. The conjecture is known to hold for the case d=n-1 (the Dembowski-Wagner theorem) and also for d=2 (a recent result established by Tonchev and the present author). Here we extend this result to the cases d=3 and d=4. The general case remains open and appears to be difficult.

Characterizing Geometric Designs II ()

We provide a characterization of the classical geometric designs formed by the points and lines of the projective space PG(n,q) of dimension n over the field with q elements, where n >= 3, among all non-symmetric (v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we also characterize the classical quasi-symmetric designs formed by the points and (n-2)-dimensional subspaces of PG(n,q), where n >= 4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and sufficiently large intersection numbers. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as the classical point-line designs; in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.

On Lander´s multiplier theorem for difference lists ()

Trace-orthogonal normal bases ()

Graphen, Netzwerke und Algorithmen ()

Effizientes Real-time-Gross-Settlement in der Wertpapierabwicklung ()

Difference sets: a second update ()

Difference set (Update) ()

Design theory ()

Difference sets: an introduction ()

Graphs, networks and algorithms ()

On small congruence covers ()

Affine TD-Ebenen ()

Affine TD-Ebenen