Geometric approach to the MacWilliams Extension Theorem for codes over modules

TL;DR

This paper extends the MacWilliams Extension Theorem to codes over modules using a geometric approach, identifying minimal code lengths for unextendable isometries and proving an extension theorem for MDS codes over modules.

Contribution

It introduces a geometric method to analyze isometry extendability for codes over modules and establishes new bounds and theorems specific to module-based codes.

Findings

Identified minimum code length for unextendable Hamming isometries in matrix modules

Proved an extension theorem for MDS codes over modules

Provided a geometric framework for code isometry analysis

Abstract

The MacWilliams Extension Theorem states that each linear Hamming isometry of a linear code extends to a monomial map. In this paper an analogue of the extension theorem for linear codes over a module alphabet is observed. A geometric approach to the extendability of isometries is described. For a matrix module alphabet we found the minimum length of a code for which an unextendable Hamming isometry exists. We also proved an extension theorem for MDS codes over a module alphabet.