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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 785–794 | Cite as

Conditions for the existence of spreads in projective Hjelmslev spaces

  • Ivan LandjevEmail author
  • Nevyana Georgieva
Article
  • 18 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

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.

Keywords

Spreads Finite chain rings Modules over finite chain rings Projective Hjelmslev spaces Galois rings 

Mathematics Subject Classification

51C05 51E23 51E05 05B25 05B30 16P10 

Notes

Acknowledgements

This result was partially supported by the Research Scientific Fund of Sofia University “St. Kl. Ohridski” under Contract No 80-10-55/19.04.2017. The authors thank the anonymous referees for the careful reading of the manuscript and for the valuable comments and suggestions.

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Authors and Affiliations

  1. 1.New Bulgarian UniversitySofiaBulgaria
  2. 2.Institute of Mathematics and InformaticsSofiaBulgaria

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