On homogeneous arcs and linear codes over finite chain rings

Abstract

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable code and the parameters of the associated arc. We prove an inequality for the largest homogeneous weight of an arc of given size N. We consider the $$\tau $$ -dual of a homogeneous arc and give an explicit formula for the homogeneous weights of the hyperplanes with respect to the dual arc. We prove that if a homogeneous arc is of constant weight, this weight is forced to be zero and the arc is a sum of neighbour classes of points. We prove various numerical conditions on the parameters of homogeneous two-weight arcs, as well as an interesting identity involving the parameters of a projective homogeneous two-weight arc and its dual, which isturns out to be also a homogeneous two-weight arc. Finally, we present examples of two-weight homogeneous arcs some of which are new.