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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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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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