Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ? → ? into a continuous planar function ?² → ?². We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.     

Arcs in the Hall Planes

Using a transformation tecnique for designs introduced in [1], I construct a class of arcs embeddeable in the Hall plane and in the dual of the Hall plane of order q proving also their completeness in the unital of Grüning. Math. Subj. Class.: 51A35 Non Desarguesian affine and projective planes. 51E22 Blocking sets, ovals, k-arcs.     

Doppelebenen und Loops

Generalizing the concept of difference sets in groups conditions are given for a loop (P, +) (with equality of left and right inverses) and a subset D of P such that (i) the left translates of D and the right translates of -D (the set of inverses for elements of D) resp. are the lines of two projective planes with point set P, forming a double plane (i.e. the lines of one plane are ovals of the other), (ii) the loop operation has a certain geometric interpretation in the double plane.

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Walter Benz zum 60. Geburtstag     

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane over a finite field F_q, where the formally defined squared Euclidean distance of every pair of points is a square in F_q. It turns out that integral point sets over F_q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case, integral point sets can be restated as cliques in Paley graphs of square order.In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over F_q for q≤47. Furthermore, we give two series of maximal integral point sets and prove their maximality.     

Sets of type (m,n) in the affine and projective planes of order nine

In this paper we give the results of exhaustive computer searches for sets of points of type (m, n) in the projective and affine planes of order nine. In particular, as the list of planes of order nine is known to be complete, our results are also complete. We also examine all known constructions of sets of type (m, n) that apply to the planes of order nine, in an attempt to summarise and extend all existing knowledge about such sets. The contrast between known constructions and our computer results leads us to conclude that sets of type (m, n) are far more numerous than was previously thought.     

Supplement to multiplier theorems

A multiplier theorem in (J. Combinatorial Theory Ser. A, in press) is extended to cyclic group divisible difference sets (GDDSs) of small size. A multiplier theorem for abelian difference sets in (Proc. Amer. Math. Soc.68 (1978), 375–379) is extended to abelian GDDSs. A remark on the existence of cyclic affine planes is made based on a previously proved multiplier theorem.     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

Duality in stable planes and related closure and kernel operations

We compare two constructions that dualize a stable plane in some sense, namely the dual plane and the opposite plane. Applying both constructions one after another we obtain a closure or kernel operation, depending on the order of execution.We examine the effect of these constructions on the automorphism group and apply our results in order to compute the automorphism groups of the complex cylinder plane, the complex united cylinder plane, and their duals. Beside the complex projective, affine, and punctured projective plane these planes are in fact the most homogeneous four-dimensional stable planes, as will be shown elsewhere [1].Supported by Studienstiftung des deutschen Volkes.     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

Degree 8 Maximal Arcs in PG(2,2), h Odd

In a recent paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes that generalised a construction of Denniston. He also gave several instances of the method to construct new maximal arcs. In this paper, the structure of the maximal arcs is examined to give geometric and algebraic methods for proving when the maximal arcs are not of Denniston type. New degree 8 maximal arcs are also constructed in PG(2,2^h), h5, h odd. This, combined with previous results, shows that every Desarguesian projective plane of (even) order greater that 8 contains a degree 8 maximal arc that is not of Denniston type.     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

Compact projective planes with homogeneous ovals

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO₂ or SO₃, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.     

Viewing sets of mutually unbiased bases as arcs in finite projective planes

This note is a short conceptual elaboration of the conjecture of Saniga et al. [J. Opt. B: Quantum Semiclass 6 (2004) L19–L20] by regarding a set of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space as an analogue of an arc in a (finite) projective plane of order d. Complete sets of MUBs thus correspond to (d + 1)-arcs, i.e., ovals. In the Desarguesian case, the existence of two principally distinct kinds of ovals for d = 2ⁿ and n  3, viz. conics and non-conics, implies the existence of two qualitatively different groups of the complete sets of MUBs for the Hilbert spaces of corresponding dimensions. A principally new class of complete sets of MUBs are those having their analogues in ovals in non-Desarguesian projective planes; the lowest dimension when this happens is d = 9.     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Some non-existence results on divisible difference sets

In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:

1. character sum arguments similar to the work of Turyn [25] for ordinary difference sets,
2. involution arguments and
3. multipliers in conjunction with results on ordinary difference sets.

Among other results, we show that an abelian affine difference set of odd orders (s not a perfect square) inG can exist only if the Sylow 2-subgroup ofG is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd ordern.     

On the original Steiner Systems

It is shown that in every n-colouring ((n ? 1)-colouring) of a projective plane (affine plane) of odd order n at least one line has three points of the same colour.     

Regular Hjelmslev planes

A projective Hjelmslev plane is called regular iff it admits an Abelian collineation group that is regular on both the points and lines of the plane and that splits into a summand regular on the elements of any given neighborhood and another summand permuting the points and lines of the projective image plane regularly. Regular Hjelmslev planes are shown to correspond to so-called special difference sets. We construct regular Hjelmslev planes with parameters (qⁿ, q) for any prime power q and any natural number n as well as for infinitely many series of parameters (t, q), where t is not a power of q. Our construction also yields series of parameters for which the existence of a Hjelmslev plane was not known up to now as well as the first information on the existence of nontrivial collineations in the case of parameters (t, q) with t not a power of q.     

Uniform Zariski’s theorem on fundamental groups

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates an isomorphism of the fundamental groups of the complements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution. The research was partially supported by an NSA grant.