A Galois Connection Approach to Wei-Type Duality Theorems

DOI10.1109/TIT.2022.3167848zbMATH Open1505.94127arXiv2011.13599OpenAlexW3109499395WikidataQ113803862 ScholiaQ113803862MaRDI QIDQ5096976

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Publication date: 19 August 2022

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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 $mathbbZ$, 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.

Full work available at URL: https://arxiv.org/abs/2011.13599

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