scroll identifier for mobile
main-content

28.08.2018

# Conditions for the existence of spreads in projective Hjelmslev spaces

Zeitschrift:
Designs, Codes and Cryptography
Autoren:
Ivan Landjev, Nevyana Georgieva
Wichtige Hinweise
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

## Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Abstract

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.

### Bitte loggen Sie sich ein, um Zugang zu diesem Inhalt zu erhalten

Literatur
Über diesen Artikel

## BranchenIndex Online

Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.

## Whitepaper

- ANZEIGE -

### Best Practices für die Mitarbeiter-Partizipation in der Produktentwicklung

Unternehmen haben das Innovationspotenzial der eigenen Mitarbeiter auch außerhalb der F&E-Abteilung erkannt. Viele Initiativen zur Partizipation scheitern in der Praxis jedoch häufig. Lesen Sie hier  - basierend auf einer qualitativ-explorativen Expertenstudie - mehr über die wesentlichen Problemfelder der mitarbeiterzentrierten Produktentwicklung und profitieren Sie von konkreten Handlungsempfehlungen aus der Praxis.