On arcs in projective Hjelmslev planes

Abstract A (k,n) -arc in the projective Hjelmslev plane PHG (R R 3 ) 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 (R R 3 ) . We present general upper bounds on the size of arcs in the projective Hjelmslev planes over chain rings R with |R|=q 2 , R/ rad R≅ F q . We summarize the known values and bounds on the cardinalities of (k,n) -arcs in the chain rings with |R|⩽25 (|R|=q 2 , R/ rad R≅ F q ) .