Characteristics of Invariant Weights Related to Code Equivalence over Rings

TL;DR

This paper investigates the algebraic conditions under which the Extension Theorem holds for weights on codes over rings, generalizing previous results and providing a framework for understanding code equivalence.

Contribution

It introduces an algebraic framework to determine when weights satisfy the Extension Theorem over rings, extending prior work on code isometries.

Findings

Established algebraic conditions for the Extension Theorem

Generalized previous results to broader classes of weights and rings

Provided a unifying framework for code equivalence analysis

Abstract

The Equivalence Theorem states that, for a given weight on the alphabet, every linear isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams' Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in [Greferath, Honold '06].