On Symmetrized Weight Compositions

TL;DR

This paper characterizes module alphabets satisfying the Extension Property for symmetrized weight compositions, establishing conditions under which modules satisfy this property, and providing a comprehensive characterization.

Contribution

It proves the necessity of embeddability in the character group for modules to satisfy the EP with respect to swc, completing the characterization.

Findings

Frobenius bimodules satisfy the EP with respect to swc

Embeddability in the character group is sufficient for EP satisfaction

Necessity of embeddability is established via a reduction to Hamming weight

Abstract

A characterization of module alphabets with the Hamming weight EP (abbreviation for Extension Property) had been settled. A thoughtfully constructed example by J.A.Wood finished the tour. Frobenius bimodules were proved to satisfy the EP with respect to symmetrized weight compositions. In 4, the embeddability in the character group of the ambient ring R was found sufficient for a module RA to satisfy the EP with respect to swc built on any subgroup of AutR(A), while the necessity remained a question. A least action trial suggests bridging to the already settled case, a trial that turns out to be successful. Here, landing in a Midway, the necessity is proved by jumping to Hamming weight. Corollary 1.11 declares a characterization of module alphabets satisfying the EP with respect to swc