Finite-Ring Combinatorics and MacWilliamsʹ Equivalence Theorem

F. J. MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been genera- lized by J. Wood who proved it for Frobenius rings using character theoretic methods. The present paper provides a combinatorial approach: First we extend I. Constantinescuʹs concept of homogeneous weights on arbitrary finite rings and prove MacWilliamsʹ equivalence theorem to hold with respect to these weights for all finite Frobenius rings. As a central tool we then establish a general inversion principle for real functions on finite modules that involves Mِbius inversion on partially ordered sets. An application of the latter yields the aforementioned result of Wood.