About the generalized Hamming weights of matrix-product codes
Rodrigo San-José
TL;DR
The paper addresses the problem of determining generalized Hamming weights (GHWs) of matrix-product codes (MPCs) by relating them to the GHWs of their constituent codes. It develops a transfer principle: sharp, computable lower bounds for $d_r(C)$ for 2×2 NSC MPCs, extends to nested NSC MPCs with $s$ constituents, and provides an accompanying upper bound that mirrors minimum-distance bounds. The results yield explicit GHW formulas for the two-constituent Reed-Solomon case and demonstrate sharpness in that setting, with further validation on Reed-Muller and related families. Together, these bounds substantially generalize known minimum-distance bounds to the full GHW hierarchy and offer practical tools for designing MPCs with controlled weight hierarchies for applications in secrecy, resilience, and network coding.
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.