  Few-weight codes over finite chain rings are associated with combinatorial
objects such as strongly regular graphs (SRGs), strongly walk-regular graphs
(SWRGs) and finite geometries, and are also widely used in data storage systems
and secret sharing schemes. The first objective of this paper is to
characterize all possible parameters of Plotkin-optimal two-homogeneous weight
regular projective codes over finite chain rings, as well as their weight
distributions. We show the existence of codes with these parameters by
constructing an infinite family of two-homogeneous weight codes. The parameters
of their Gray images have the same weight distribution as that of the
two-weight codes of type SU1 in the sense of Calderbank and Kantor (Bull Lond
Math Soc 18: 97-122, 1986). Further, we also construct three-homogeneous weight
regular projective codes over finite chain rings combined with some known
results. Finally, we study applications of our constructed codes in secret
sharing schemes and graph theory. In particular, infinite families of SRGs and
SWRGs with non-trivial parameters are obtained.
