On the Weight Distribution of Codes over Finite Rings

Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.