Simple proofs for duality of generalized minimum poset weights and weight distributions of (Near-)MDS poset codes

TL;DR

This paper provides simple matroid-based proofs for the duality of generalized minimum poset weights and derives their weight distributions for MDS and Near-MDS poset codes, enhancing theoretical understanding.

Contribution

The paper introduces a straightforward matroid-theoretic proof of duality for generalized minimum poset weights and determines their weight distributions for MDS and Near-MDS codes.

Findings

Matroid theory simplifies the proof of duality.

Derived weight distributions for MDS and Near-MDS poset codes.

Clarified the relation between rank and corank functions in this context.

Abstract

In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Recently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-MDS poset codes in the same spirit.