About the generalized Hamming weights of matrix-product codes

TL;DR

This paper establishes bounds and explicit formulas for the generalized Hamming weights of matrix-product codes, especially focusing on cases with Reed-Solomon constituent codes and non-nested configurations.

Contribution

It introduces new bounds and explicit formulas for generalized Hamming weights of matrix-product codes, including non-nested cases and Reed-Solomon codes.

Findings

Derived a general lower bound for nested matrix-product codes.

Provided an upper bound similar to minimum distance bounds.

Obtained explicit formulas for Reed-Solomon constituent codes.

Abstract

We derive a general lower bound for the generalized Hamming weights of nested matrix-product codes, with a particular emphasis on the cases with two and three constituent codes. We also provide an upper bound which is reminiscent of the bounds used for the minimum distance of matrix-product codes. When the constituent codes are two Reed-Solomon codes, we obtain an explicit formula for the generalized Hamming weights of the resulting matrix-product code. We also deal with the non-nested case for the case of two constituent codes.