Scattered subspaces and related codes

TL;DR

This paper extends the connection between scattered subspaces and MRD codes to all intermediate dimensions, providing new examples of non-square MRD codes and classifying codes with related parameters.

Contribution

It generalizes the known links between scattered subspaces and MRD codes for all h, introduces new non-square MRD codes, and classifies codes with similar parameters.

Findings

Established a unified connection for all h between scattered subspaces and MRD codes.

Provided examples of non-square MRD codes not equivalent to known classes.

Determined the weight distribution of related codes.

Abstract

After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we propose a connection for any $h\in\{1,\ldots,r-1\}$, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum $h$-scattered subspaces.