A Galois Connection Approach to Wei-Type Duality Theorems

TL;DR

This paper introduces a new Galois connection perspective to unify and extend Wei-type duality theorems, connecting generalized Hamming weights and code profiles through a broad duality framework.

Contribution

It presents a general Wei-type duality theorem based on Galois connections, unifying previous results and deriving new dualities for $w$-demimatroids and $w$-demi-polymatroids.

Findings

Unified Wei-type duality theorem for Galois connections

New duality results for $w$-demimatroids and $w$-demi-polymatroids

Generalization of known duality theorems across various metrics

Abstract

In $1991$, Wei proved a duality theorem that established an interesting connection between the generalized Hamming weights of a linear code and those of its dual code. Wei's duality theorem has since been extensively studied from different perspectives and extended to other settings. In this paper, we re-examine Wei's duality theorem and its various extensions, henceforth referred to as Wei-type duality theorems, from a new Galois connection perspective. Our approach is based on the observation that the generalized Hamming weights and the dimension/length profiles of a linear code form a Galois connection. The central result in this paper is a general Wei-type duality theorem for two Galois connections between finite subsets of $\mathbb{Z}$, from which all the known Wei-type duality theorems can be recovered. As corollaries of our central result, we prove new Wei-type duality theorems for $w$-demimatroids defined over finite sets and $w$-demi-polymatroids defined over modules with a composition series, which further allows us to unify and generalize all the known Wei-type duality theorems established for codes endowed with various metrics.