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A Galois Connection Approach to Wei-Type Duality Theorems.
- Source :
-
IEEE Transactions on Information Theory . Aug2022, Vol. 68 Issue 8, p5133-5144. 12p. - Publication Year :
- 2022
-
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 of 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 $w$ -demi-matroids 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LINEAR codes
*HAMMING weight
*MATROIDS
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 68
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 157958043
- Full Text :
- https://doi.org/10.1109/TIT.2022.3167848