MacWilliams extension theorems and the local–global property for codes over Frobenius rings

The MacWilliams extension theorem is investigated for various weight functions over finite Frobenius rings. The problem is reformulated in terms of a local–global property for subgroups of the general linear group. Among other things, it is shown that the extension theorem holds true for poset weights if and only if the underlying poset is hierarchical. Specifically, the Rosenbloom–Tsfasman weight for vector codes satisfies the extension theorem, whereas the Niederreiter–Rosenbloom–Tsfasman weight for matrix codes does not. A short character-theoretic proof of the well-known MacWilliams extension theorem for the homogeneous weight is provided. Moreover it is shown that the extension theorem carries over to direct products of weights, but not to symmetrized products.