Evasive subspaces, generalized rank weights and near MRD codes
Abstract
We revisit and extend the connections between $\mathbb{F}_{q^m}$linear rankmetric 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 rankmetric 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 quasiMRD codes with $h$scattered subspaces of maximum dimension, extending to all the parameters sets the already known results on MRD codes.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 arXiv:
 arXiv:2204.11791
 Bibcode:
 2022arXiv220411791M
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Information Theory;
 94B05;
 51E20;
 94B27;
 11T71
 EPrint:
 20 pages