Generalizations of Wei's Duality Theorem

TL;DR

This paper extends Wei's Duality Theorem to new combinatorial structures called demi-matroids, providing a unified framework for duality in codes, matroids, and graph theory with new proofs and generalizations.

Contribution

It introduces demi-matroids as a new combinatorial object and establishes duality theorems for them, generalizing Wei's Duality Theorem and related results.

Findings

Demi-matroids generalize matroids and describe duality in Wei's theorem.

New duality theorems for codes, matroids, and graphs are established.

A combinatorial proof of Wei's Duality Theorem is provided.

Abstract

Wei's celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two Wei-type duality theorems for new combinatorial structures that are introduced and named {\em demi-matroids}. These generalize matroids and are the appropriate combinatorial objects for describing the duality in Wei's Duality Theorem. A new proof of the Duality Theorem is thereby given that explains the theorem in combinatorial terms. Special cases of the general duality theorems are also given, including duality theorems for cycles and bonds in graphs and for transversals.