A remark on ${\mathbb F}_{q^n}$-Linear MRD codes

TL;DR

This paper characterizes elements of minimum rank in generalized Gabidulin codes using Grassmann coordinates, providing new insights into the structure of MRD-codes and linearized polynomials with specific rank constraints.

Contribution

It introduces a novel description of minimum rank elements in generalized Gabidulin codes and derives parametric equations for MRD-codes of a given distance.

Findings

Characterization of minimum rank elements via Grassmann coordinates

Parametric equations for MRD-codes with distance d=n-k+1

New insights into linearized polynomials of rank at most n-k

Abstract

In this note, we provide a description of the elements of minimum rank of a generalized Gabidulin code in terms of Grassmann coordinates. As a consequence, a characterization of linearized polynomials of rank at most $n-k$ is obtained, as well as parametric equations for MRD-codes of distance $d=n-k+1$.