A new family of $2$-scattered subspaces and related MRD codes

TL;DR

The paper proves the existence of maximum $2$-scattered $\mathbb{F}_q$-subspaces in $V(r,q^6)$ for all $r\ge 3$, $r\ne 5$ when $q=2^h$ with $h$ odd, via an explicit construction $U$ in $\mathbb{F}_{q^6}^4$ and a comprehensive case analysis ensuring every $2$-dimensional subspace intersects $U$ in at most $q^2$-dimensional $\mathbb{F}_q$-subspaces, i.e., $U$ is $2$-scattered. The construction ties to MRD codes: the associated rank-metric code is $2$-MRD (with $d_2=n-k+2$) but not $1$-MRD, and thus demonstrates the existence of $2$-MRD codes that are not $1$-MRD. The work extends the catalogue of maximum $h$-scattered subspaces and clarifies connections between finite geometry and rank-metric coding theory, including related dualities and generalized weights. Overall, it delivers a new infinite family of geometric objects and their MRD-code counterparts with potential applications in network coding and design theory.

Abstract

Scattered subspaces and $h$-scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While numerous constructions of scattered subspaces exist, relatively few are known about $h$-scattered subspaces with $h\geq2$. In this paper, we establish the existence of maximum $2$-scattered $\F_q$-subspaces in $V(r,q^6)$ whenever $r\geq 3$, $r\ne 5$, and $q$ is an odd power of $2$. Additionally, we explore the corresponding MRD codes.

Theorems & Definitions (38)

• Definition 1.1
• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• proof
• Theorem 3.1