Evasive subspaces, generalized rank weights and near MRD codes

TL;DR

This paper explores the geometric relationship between rank-metric codes and evasive subspaces, providing new insights into generalized rank weights, near MRD codes, and quasi-MRD codes through a unified geometric framework.

Contribution

It introduces a unifying geometric framework linking rank-metric codes with evasive subspaces, simplifying proofs and characterizing near MRD and quasi-MRD codes.

Findings

Provided elementary proofs of known results on scattered subspaces

Characterized near MRD codes via geometric properties

Extended results on MRD codes to all parameter sets

Abstract

We revisit and extend the connections between $\mathbb{F}_{q^m}$-linear rank-metric codes and evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We give a unifying framework in which we prove in an elementary way how the parameters of a rank-metric code are related to special geometric properties of the associated evasive subspace, with a particular focus on the generalized rank weights. In this way, we can also provide alternative and very short proofs of known results on scattered subspaces. We then use this simplified point of view in order to get a geometric characterization of near MRD codes and a clear bound on their maximal length. Finally we connect the theory of quasi-MRD codes with $h$-scattered subspaces of maximum dimension, extending to all the parameters sets the already known results on MRD codes.