#### Vol. 4, No. 1, 2006

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Caps in projective Hjelmslev spaces over finite chain rings of nilpotency index $2$

### Thomas Honold and Ivan N. Landjev

Vol. 4 (2006), No. 1, 13–25
##### Abstract

We investigate caps in the projective Hjelmslev geometries PHG$\left({R}_{R}^{k}\right)$ over chain rings $R$ with $|R|={q}^{2}$, $R∕radR\cong {\mathbb{F}}_{q}$. We present a geometric construction for caps using ovoids in the factor geometry $PG\left(3,q\right)$ as well as an algebraic construction that makes use of the Teichmüller group of units in the Galois extension of certain chain rings. We prove upper bounds on the size of a maximal cap in PHG$\left({R}_{R}^{4}\right)$. It has an order of magnitude ${q}^{4}$. This bound extends to higher dimensions, but gives the rather rough estimate ${q}^{2k-4}$.

##### Mathematical Subject Classification 2000
Primary: 51E21, 51E22, 51E26, 94B05
##### Milestones
Received: 8 April 2006
Accepted: 23 October 2006
##### Authors
 Thomas Honold Ivan N. Landjev