MacWilliams Extension Theorems and the Local-Global Property for Codes over Rings

TL;DR

This paper explores the conditions under which the MacWilliams extension theorem holds for various weights over finite Frobenius rings, linking it to a local-global property and analyzing specific weights like poset and Rosenbloom-Tsfasman.

Contribution

It characterizes when the extension theorem applies to different weights and introduces a local-global perspective, including new proofs and results for product weights.

Findings

Extension theorem holds for hierarchical poset weights.

Rosenbloom-Tsfasman weight satisfies the extension theorem.

Extension theorem applies to direct products of weights, but not to symmetrized products.

Abstract

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.