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

Abstract

We investigate caps in the projective Hjelmslev geometries PHG$\left(R_R^k\right)$ over chain rings $R$ with $\left|R\right|=q^2$, $R∕radR\cong {\mathbb{F}}_q$. 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 $q^{2k-4}$.