Generalising the scattered property of subspaces
Bence Csajbók, Giuseppe Marino, Olga Polverino, Ferdinando Zullo
TL;DR
The paper generalizes the scattered subspace notion to $h$-scattered subspaces over ${\mathbb F}_q$ and establishes a sharp dimension bound $\dim_{\mathbb F_q} U \le \frac{rn}{h+1}$ for $h>1$, with equality achievable when $h+1$ divides $r$. It develops a Delsarte duality linking maximum $h$-scattered subspaces to maximum $(n-h-2)$-scattered duals and connects these structures to MRD-codes via left/right idealisers. The study then translates these geometric objects into ${\mathbb F}_q$-linear sets, showing that for $h\ge 2$ linear-set equivalence is governed by $\mathrm{GL}(r,q^n)$-equivalence of the defining subspaces, and explores the MRD-code correspondence in this broader setting. Finally, the authors derive sharp intersection bounds with hyperplanes by a rigorous double-counting argument involving Gaussian binomial identities, enriching the interplay between subspace geometry, linear sets, and rank-metric codes.
Abstract
Let $V$ be an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. We call an $\mathbb{F}_q$-subspace $U$ of $V$ $h$-scattered if $U$ meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $h$. In 2000 Blokhuis and Lavrauw proved that $\dim_{\mathbb{F}_q} U \leq rn/2$ when $U$ is $1$-scattered. Subspaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to $rn/2$-dimensional $1$-scattered subspaces and to $n$-dimensional $(r-1)$-scattered subspaces. In this paper we prove the upper bound $rn/(h+1)$ for the dimension of $h$-scattered subspaces, $h>1$, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.