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

A Galois Connection Approach to Wei-Type Duality Theorems