Designs in projective Hjelmslev spaces

Titelangaben

Kiermaier, Michael ; Landjev, Ivan:
Designs in projective Hjelmslev spaces.
In: Lavrauw, Michel (Hrsg.): Theory and Applications of Finite Fields. - Providence, RI : American Mathematical Society , 2012 . - S. 111-121 . - (Contemporary Mathematics ; 579 )
ISBN 978-0-8218-5298-9
DOI: https://doi.org/10.1090/conm/579/11524

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Abstract

In this paper, we define analogs of designs in the Grassmanian of all submodules of given shape in the free module Rⁿ, where R is a finite chain ring. We consider τ-designs for τ=(m,0,...,0). Such designs are equivalent to spreads in projective Hjelmslev geometries. We give examples of line spreads in which no two lines are neighbours in the 3-dimensional Hjelmslev geometries over the chain rings with four and nine elements. We prove that non-trivial line spreads exist in all 3-dimensional projective Hjelmslev geometries over a commutative chain ring R. We give a necessary and sufficient condition for the existence of spreads of Hjelmslev subspaces. We prove that in case of spreads of non-free subspaces, this necessary condition is not sufficient.