New MRD codes from linear cutting blocking sets

TL;DR

This paper introduces a new family of maximum rank distance (MRD) codes with improved parameters, constructed via linear cutting blocking sets, and demonstrates they are not decomposable into smaller MRD codes.

Contribution

First construction of a family of $ ext{F}_{q^m}$-linear MRD codes of length 2m with larger generalized rank weights, not formed as a direct sum of smaller codes.

Findings

New family of MRD codes with better parameters

Codes have strictly larger generalized rank weights

Demonstrates not all MRD codes share the same generalized rank weights

Abstract

Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated $q$-system. Using this result, we provide the first construction of a family of $\mathbb{F}_{q^m}$-linear MRD codes of length $2m$ that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting.