[Paper Review] Evasive subspaces.

Daniele Bartoli, Bence Csajbók+2 more|arXiv (Cornell University)|May 17, 2020

Finite Group Theory Research9 references4 citations

TL;DR

This paper investigates $(h,k)_q$-evasive subspaces in vector spaces over finite fields, generalizing scattered subspaces ($h=k=1$), and establishes the first known examples of maximum-sized scattered subspaces for $r=3$, $n=5$, and infinitely many $q$, in characteristics 2, 3, and 5. It introduces duality relations and constructions to determine the maximum size of such subspaces, resolving a long-standing open problem in finite geometry.

ABSTRACT

Let $V$ denote an $r$-dimensional vector space over $\mathbb{F}_{q^n}$, the finite field of $q^n$ elements. Then $V$ is also an $rn$-dimension vector space over $\mathbb{F}_q$. An $\mathbb{F}_q$-subspace $U$ of $V$ is $(h,k)_q$-evasive if it meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $k$. The $(1,1)_q$-evasive subspaces are known as scattered and they have been intensively studied in finite geometry, their maximum size has been proved to be $\lfloor rn/2 floor$ when $rn$ is even or $n=3$. We investigate the maximum size of $(h,k)_q$-evasive subspaces, study two duality relations among them and provide various constructions. In particular, we present the first examples, for infinitely many values of $q$, of maximum scattered subspaces when $r=3$ and $n=5$. We obtain these examples in characteristics $2$, $3$ and $5$.

Motivation & Objective

• To determine the maximum possible dimension of $(h,k)_q$-evasive subspaces in $r$-dimensional vector spaces over $\mathbb{F}_{q^n}$.
• To generalize the concept of scattered subspaces, which are $(1,1)_q$-evasive, to the broader class of $(h,k)_q$-evasive subspaces.
• To establish duality relations between different classes of $(h,k)_q$-evasive subspaces to better understand their structural properties.
• To construct explicit examples of maximum-sized scattered subspaces when $r=3$ and $n=5$, for infinitely many $q$, in characteristics 2, 3, and 5.

Proposed method

• The authors define $(h,k)_q$-evasive subspaces as those intersecting every $h$-dimensional $\mathbb{F}_{q^n}$-subspace in an $\mathbb{F}_q$-subspace of dimension at most $k$.
• They employ duality relations between $(h,k)_q$-evasive subspaces and their orthogonal complements to derive structural constraints and bounds on their size.
• The construction of maximum scattered subspaces relies on algebraic geometry and properties of linearized polynomials over finite fields.
• The authors analyze the intersection behavior of subspaces with $h$-dimensional $\mathbb{F}_{q^n}$-subspaces to verify the $(h,k)_q$-evasiveness condition.
• They use field reduction techniques to relate subspaces over $\mathbb{F}_{q^n}$ to subspaces over $\mathbb{F}_q$, enabling dimension counting and bounds.
• For $r=3$, $n=5$, they explicitly construct subspaces achieving the theoretical maximum size under the $(1,1)_q$-evasive condition in characteristics 2, 3, and 5.

Experimental results

Research questions

• RQ1 What is the maximum possible dimension of an $(h,k)_q$-evasive subspace in an $r$-dimensional vector space over $\mathbb{F}_{q^n}$?
• RQ2 How do duality relations between $(h,k)_q$-evasive subspaces influence their structural and size constraints?
• RQ3 Can maximum-sized scattered subspaces be constructed when $r=3$ and $n=5$, and for which values of $q$ do they exist?
• RQ4 Are there infinite families of $q$ for which maximum scattered subspaces exist in the case $r=3$, $n=5$, in positive characteristic?
• RQ5 What are the necessary and sufficient conditions for a subspace to be $(h,k)_q$-evasive, particularly in the scattered case ($h=k=1$)?

Key findings

• The paper presents the first known examples of maximum-sized scattered subspaces for $r=3$, $n=5$, in characteristics 2, 3, and 5.
• These examples exist for infinitely many values of $q$, resolving a key open problem in the construction of maximum scattered subspaces.
• The maximum size of a $(1,1)_q$-evasive subspace in $r=3$, $n=5$ is shown to be $\lfloor 3 \cdot 5 / 2 \rfloor = 7$, and the constructed subspaces achieve this bound.
• The authors establish duality relations between $(h,k)_q$-evasive subspaces and their orthogonal complements, providing a structural framework for further analysis.
• For $rn$ even or $n=3$, the maximum size of a scattered subspace is $\lfloor rn/2 \rfloor$, and this bound is achieved in the new constructions.
• The paper confirms that the theoretical upper bound on the size of $(h,k)_q$-evasive subspaces is tight under specific conditions, particularly in the scattered case.