The maximum number of homogeneous weights of linear codes over chain rings

The problem of determining the largest possible number of distinct Hamming weights in several classes of codes over finite fields was studied recently in several papers (Shi et al. in Des Codes Cryptogr 87(1):87–95, 2019, in IEEE Trans Inf Theory 66(11):6855–6862, 2020; Chen et al. in IEEE Trans Inf Theory 69(2):995–1004, 2022). A further problem is to find the minimum length of codes meeting those bounds with equality. These two questions are extended here to linear codes over chain rings for the homogeneous weight. An explicit upper bound is given for codes of given type and arbitrary length as a function of the residue field size. This bound is then shown to be tight by an argument based on Hjemslev geometries. The second question is studied for chain rings with residue field of order two.