Hjelmslev Geometries Research Articles

On homogeneous arcs and linear codes over finite chain rings

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable code and the parameters of the associated arc. We prove an inequality for the largest homogeneous weight of an arc of given size N. We consider the $$\tau $$ -dual of a homogeneous arc and give an explicit formula for the homogeneous weights of the hyperplanes with respect to the dual arc. We prove that if a homogeneous arc is of constant weight, this weight is forced to be zero and the arc is a sum of neighbour classes of points. We prove various numerical conditions on the parameters of homogeneous two-weight arcs, as well as an interesting identity involving the parameters of a projective homogeneous two-weight arc and its dual, which isturns out to be also a homogeneous two-weight arc. Finally, we present examples of two-weight homogeneous arcs some of which are new.

Hjelmslev's geometry of reality

During the first half of the 20th century the Danish geometer Johannes Hjelmslev developed what he called a geometry of reality. It was presented as an alternative to the idealized Euclidean paradigm that had recently been completed by Hilbert. Hjelmslev argued that his geometry of reality was superior to the Euclidean geometry both didactically, scientifically and in practice: Didactically, because it was closer to experience and intuition, in practice because it was in accordance with the real geometrical drawing practice of the engineer, and scientifically because it was based on a smaller axiomatic basis than Hilbertian Euclidean geometry but still included the important theorems of ordinary geometry. In this paper, I shall primarily analyze the scientific aspect of Hjelmslev's new approach to geometry that gave rise to the so-called Hjelmslev (incidence) geometry or ring geometry.

The Thomsen–Bachmann correspondence in metric geometry II

We continue the investigations of the Thomsen–Bachmann correspondence between metric geometries and groups, which is often summarized by the phrase ‘Geometry can be formulated in the group of motions’. In the first part (H. Struve and R. Struve in J Geom, 2019. https://doi.org/10.1007/s00022-018-0465-8 ) of this paper it was shown that the Thomsen–Bachmann correspondence can be precisely stated in a framework of first-order logic. We now prove that the correspondence, which was established by Thomsen and Bachmann for Euclidean and for plane absolute geometry, holds also for Hjelmslev geometries, Cayley–Klein geometries, isotropic and equiform geometries, and that these geometries and the theory of their group of motions are mutually faithfully interpretable (and bi-interpretable, but not definitionally equivalent). Hence a reflection-geometric axiomatization of a class of motion groups corresponds to an elementary axiomatization of the underlying geometry and provides with the calculus of reflections a powerful proof method.

Conditions for the existence of spreads in projective Hjelmslev spaces

Let R be a finite chain ring with $$|R|=q^m$$ , and $$R/\text {Rad }R\cong \mathbb {F}_q$$ . Denote by $$\varPi ={{\mathrm{PHG}}}({}_RR^n)$$ the (left) $$(n-1)$$ -dimensional projective Hjelmslev geometry over R. As in the classical case, we define a $$\lambda $$ -spread of $$\varPi $$ to be a partition of its pointset into subspaces of shape $$\lambda =(\lambda _1,\ldots ,\lambda _n)$$ . An obvious necessary condition for the existence of a $$\lambda $$ -spread $$\mathcal {S}$$ in $$\varPi $$ is that the number of points in a subspace of shape $$\lambda $$ divides the number of points in $$\varPi $$ . If the elements of $$\mathcal {S}$$ are Hjelmslev subspaces, i.e., free submodules of $${}_RR^n$$ , this necessary condition is also sufficient. If the subspaces in $$\mathcal {S}$$ are not Hjelmslev subspaces this numerical condition is not sufficient anymore. For instance, for chain rings with $$m=2$$ , there is no spread of shape $$\lambda =(2,2,1,0)$$ in $${{\mathrm{PHG}}}({}_RR^4)$$ . An important (and maybe difficult) question is to find all shapes $$\lambda $$ , for which $$\varPi $$ has a $$\lambda $$ -spread. In this paper, we present a construction which gives spreads by subspaces that are not necessarily Hjelmslev subspaces. We prove the non-existence of spreads of shape $$2^{n/2}1^a$$ [cf. (2)], $$1\le a\le n/2-1$$ , in $${{\mathrm{PHG}}}({}_RR^n)$$ , where n is even and R is a chain ring of length 2.

Complex Curves as Lines of Geometries

We investigate Hjelmslev geometries \({\mathcal{H}}\) having a representation in a complex affine space \({\mathbb{C}^n}\) the lines of which are given by entire functions. If \({\mathcal{H}}\) has dimension 2 and the entire functions satisfy some injectivity conditions, then \({\mathcal{H}}\) is a substructure of the complex Laguerre plane. If the lines are geodesics with respect to a natural connection \({\nabla^{\circ}}\), then a detailed classification of them as well as of the corresponding geometries is obtained. Generalizations of complex Grunwald planes play a main role in the classification. Since in the considered geometries the set of lines is invariant under the translation group of \({\mathbb{C}^n}\), we classify all complex curves C in \({\mathbb{C}^n}\) given by entire functions as well as the connections \({\nabla^\circ}\) such that all images of C under the translation group of \({\mathbb C^n}\) consist of geodesics with respect to \({\nabla^{\circ}}\).

Imbrex geometries

We introduce an axiom on strong parapolar spaces of diameter 2, which arises naturally in the framework of Hjelmslev geometries. This way, we characterize the Hjelmslev–Moufang plane and its relatives (line Grassmannians, certain half-spin geometries and Segre geometries). At the same time we provide a more general framework for a lemma of Cohen, which is widely used to study parapolar spaces. As an application, if the geometries are embedded in a projective space, we provide a common characterization of (projections of) Segre varieties, line Grassmann varieties, half-spin varieties of low rank, and the exceptional variety E6,1 by means of a local condition on tangent spaces.

Codes from translation schemes on Galois rings of characteristic 4

Abstract In [Thomas Honold. Two-intersection sets in projective Hjelmslev spaces. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pages 1807–1813, 2010], it has been shown that the Teichmuller point set in the projective Hjelmslev geometry PHG ( R k ) over a Galois ring R of characteristic 4 with k odd is a two-intersection set. From this result, the parameters of the generated codes can be derived, see [Michael Kiermaier and Johannes Zwanzger. New ring-linear codes from dualization in projective Hjelmslev geometries. To appear in Des. Codes Cryptogr. doi: 10.1007/s10623-012-9650-1 , Fact 5.2]. The resulting Teichmuller Codes have a high minimum distance. The key step in the proof of the two-weight property in [Thomas Honold. Two-intersection sets in projective Hjelmslev spaces. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, pages 1807–1813, 2010.] is to show that for a certain supergroup Σ of the Teichmuller units T in a Galois ring S of characteristic 4, the partition A Σ = { { 0 } , 2 S ⁎ , Σ , S ⁎ Σ } induces a translation scheme on ( S , + ) . We generalize these results by characterizing all supergroups Σ of T such that A Σ induces a symmetric translation scheme. In turn, we get new two-intersection sets in projective Hjelmslev geometries and two new series T q , k , s and U q , k , s of R-linear codes. The series T q , k , s generalizes the Teichmuller codes (special case s = 0 ). The codes U q , k , s are homogeneous two-weight codes. Application of the dualization construction to T q , k , s yields another series T q , k , s ⁎ . The Gray images of the codes T q , k , s and T q , k , s ⁎ have a higher minimum distance than all known F q -linear codes of the same length and size.

New ring-linear codes from dualization in projective Hjelmslev geometries

In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include $${\mathbb {Z}_4}$$ as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. Applying it to the $${\mathbb {Z}_4}$$ -preimages of the Kerdock codes and a related family of codes we will call Teichmuller codes, we get two new infinite series of codes and compute their symmetrized weight enumerators. In some cases, residuals of the original code give further interesting codes. The generalized Gray map translates our codes into ordinary, generally non-linear codes in the Hamming space. The obtained parameters include (58, 27, 28)2, (60, 28, 28)2, (114, 28, 56)2, (372, 210, 184)2 and (1988, 212, 992)2 which provably have higher minimum distance than any linear code of equal length and cardinality over an alphabet of the same size (better-than-linear, BTL), as well as (180, 29, 88)2, (244, 29, 120)2, (484, 210, 240)2 and (504, 46, 376)4 where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).

Non-free extensions of the simplex codes over a chain ring with four elements

Let R be a chain ring with four elements. In this paper, we present two new constructions of R-linear codes that contain a subcode associated with a simplex code over the ring R. The simplex codes are defined as the codes generated by a matrix having as columns the homogeneous coordinates of all points in some projective Hjelmslev geometry PHG(R k ). The first construction generalizes a recent result by Kiermaier and Zwanzger to codes of arbitrary dimension. We provide a geometric interpretation of their construction which is then extended to projective Hjelmslev spaces of arbitrary dimension. The second construction exploits the possibility of adding two non-free rows to the generator matrix of a linear code over R associated with a given point set. Though the construction works over both chain rings with four elements, the better codes are obtained for $${R=\mathbb{Z}_4}$$ .

The dual construction for arcs in projective Hjelmslev spaces

In this paper, we present a duality construction for multiarcs in projective Hjelmslev geometries over chain rings of nilpotency index 2. We compute the parameters of the resulting arcs and discuss some examples.

Hjelmslev geometry of mutually unbiased bases

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p2 and rank r. The q vectors of a basis of correspond to the q points of a (so-called) neighbour class and the q + 1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.

Linear Codes over Finite Chain Rings

The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a well-known result for linear codes over fields, we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa. Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings.