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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”.

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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.

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