Related papers: Evasive subspaces

We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a…

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. In this paper we describe the structure of maximal non-trivial $t$-intersecting families of $k$-dimensional subspaces of $V$ with large size. We also determine…

Let $F$ be a field. A $2$-$(7, 3, 1)_F$-subspace design, or $q$-Fano plane, over $F$, is a $7$-dimensional vector space $V$ over $F$ together with a collection $\mathfrak{B}$ of three-dimensional subspaces of $V$ such that every…

Let $V=V(n,q)$ denote the vector space of dimension $n$ over the finite field with $q$ elements. A subspace partition ${\mathcal P}$ of $V$ is a collection of nontrivial subspaces of $V$ such that each nonzero vector of $V$ is in exactly…

Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric…

In this paper we show that the largest possible size of a subset of $\mathbb{F}_q^n$ avoiding right angles, that is, distinct vectors $x,y,z$ such that $x-z$ and $y-z$ are perpendicular to each other is at most $O(n^{q-2})$. This improves…

Given an irreducible representation of $SL_2(F_q)$ for an odd prime $q\geq 5$, we find the dimension of the space of cusp forms with respect to the full modular group taking values in the representation space. The dimension equals the…

A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies…

We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered…

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum…

Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, and $\mathcal{F}$ be a field. A vector $u\in\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ is called a product vector if $u=u^{[1]}\otimes\cdots\otimes u^{[n]}$ for some…

We find the exact size of a maximal non-commuting set in unipotent uppertriangular linear group $UU_4(\mathbb{F}_q)$ in terms of a non-commuting geometric structure (Refer Definition [10]), where $\mathbb{F}_q$ is the finite field with $q$…

Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting…

We introduce and study the spherical dimension, a natural topological relaxation of the VC dimension that unifies several results in learning theory where topology plays a key role in the proofs. The spherical dimension is defined by…

For any set system $H=(V,R), \ R \subseteq 2^V$, a subset $S \subseteq V$ is called \emph{shattered} if every $S' \subseteq S$ results from the intersection of $S$ with some set in $\R$. The \emph{VC-dimension} of $H$ is the size of a…

We introduce a novel concept of rank for subsets of finite metric spaces E^n_q (the set of all n-dimensional vectors over an alphabet of size q) equipped with the Hamming distance, where the rank R(A) of a subset A is defined as the number…

Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that…

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices…

It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$…

We denote by $\mathcal{P}_q$ the vector space of functions from a finite field $\mathbb{F}_q$ to itself, which can be represented as the space $\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)$ of polynomial functions. We denote by $\mathcal{O}_n…