Arcs in projective Hjelmslev planes

The $(k,n)$-arcs in projective Hjelmslev plane $\mathrm{PHG}(R_R^3)$ over a finite chain ring $R$ are considered. We prove general upper bounds on the cardinality of such arcs and establish the maximum possible size of the projective $(k,n)$-arcs with $n\in\{q^2,\ldots,q^2+q-1\}$. Constructions of projective arcs in the Hjelmslev planes over the chain rings with 4 and 9 elements are also given.