A new infinite family of maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(m(h+1),q^n)$ and associated MRD codes

TL;DR

This paper introduces an infinite family of maximum $h$-scattered $F_q$-subspaces in finite vector spaces and constructs associated MRD codes with improved generalized weights, advancing finite geometry and coding theory.

Contribution

It presents a new infinite family of $h$-subspaces and their MRD codes, with analysis of their generalized weights, showing improvements over existing codes.

Findings

New infinite family of $h$-subspaces constructed.

Associated MRD codes have larger generalized weights.

Enhanced understanding of the structure and properties of these codes.

Abstract

The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces play a pivotal role in analyzing various geometric structures such as blocking sets, two-intersection sets, complete arcs, caps in affine and projective spaces over finite fields and rank metric codes. This paper introduces a new infinite family of $h$-subspaces, along with their associated MRD codes. Additionally, it addresses the task of determining the generalized weights of these codes. Notably, we demonstrate that these MRD codes exhibit some larger generalized weights compared to those previously identified.