On arcs in projective Hjelmslev planes Original Research Article

A (k,n)-arc in the projective Hjelmslev plane PHG(RR3) is defined as a set of k points in the plane such that some n but no n+1 of them are collinear. In this paper, we consider the problem of finding the largest possible size of a (k,n)-arc in PHG(RR3). We present general upper bounds on the size of arcs in the projective Hjelmslev planes over chain rings R with |R|=q2, R/rad R≅Fq. We summarize the known values and bounds on the cardinalities of (k,n)-arcs in the chain rings with |R|⩽25 (|R|=q2, R/rad R≅Fq).