Finite-ring combinatorics and MacWilliams' equivalence theorem

F.J. MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been generalized by J. Wood who proved it For Frobenius rings using character theoretic method s. The present paper provides a combinatorial approach: First we extend I. Constantinescu's concept of homogeneous weights on arbitrary finite rings a nd prove MacWilliams' equivalence theorem to hold with respect to these wei ghts for all finite Frobenius rings. As a central tool we then establish a general inversion principle for real functions on finite modules that invol ves Mobius inversion on partially ordered sets. An application of the latte r yields the aforementioned result of Wood. (C) 2000 Academic Press.