Maximum scattered subspaces are not only objects of intrinsic interest in finite geometry but also powerful tools for the construction of MRD-codes, projective two-weight codes, and strongly regular graphs. In 2018 Csajbók, Marino, Polverino, and Zanella introduced a new family of maximum scattered subspaces in $\mathbb{F}_{q^6} \times \mathbb{F}_{q^6}$ arising from polynomials of type $f_b(x)=bx^q+x^{q^4}$ for certain choices of $b \in \mathbb{F}_{q^6}$. Throughout characterizations for $f_{b_2}(x)$ and $f_{b_1}(x)$ giving rise to equivalent maximum scattered subspaces, the authors conjectured that the portion of new and inequivalent maximum scattered subspaces obtained in this way is quite large. In this paper first we find necessary and sufficient conditions for $b$ to obtain a maximum scattered subspace. Such conditions were found independently with different techniques also by Polverino and Zullo 2019. Then we prove the conjecture on the number of new and inequivalent maximum scattered subspaces of this family.
Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the $\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for infinitely many $m$. Our main results are the complete classification…
After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we propose a connection for any $h\in\{1,\ldots,r-1\}$, extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. Up to equivalence, we classify MRD codes having the same parameters as the ones in our connection. Also, we deter…
Maximum 2-scattered subspaces of $V(r,q^6)$ for every $r\geq 3$
Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze（佛罗伦萨大学）；Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）
A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.
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 $\lfloor rn/2 \rfloor$ when $rn$ is even or $n=3$. We investigate the maximum size of $…
We answer a question by Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields by proving a conjecture by Ghorpade and Ram.
Let $f(X) \in \mathbb{F}_{q^r}[X]$ be a $q$-polynomial. If the $\mathbb{F}_q$-subspace $U=\{(x^{q^t},f(x)) \mid x \in \mathbb{F}_{q^n}\}$ defines a maximum scattered linear set, then we call $f(X)$ a scattered polynomial of index $t$. The asymptotic behaviour of scattered polynomials of index $t$ is an interesting open problem. In this sense, exceptional scattered polynomials of index $t$ are those for which $U$ is a maximum scattered linear set in ${\rm PG}(1,q^{mr})$ for infinitely many $m$. The complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and …
Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. Define a real-valued weight function on the $1$-dimensional vector spaces of $V$ such that the sum of all weights is zero. Let the weight of a subspace $S$ be the sum of the weights of the $1$-dimensional subspaces contained in $S$. In 1988 Manickam and Singhi conjectured that if $n \geq 4k$, then the number of $k$-dimensional subspaces with nonnegative weight is at least the number of $k$-dimensional subspaces on a fixed $1$-dimensional subspace. Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the…
In [2] and [19] are presented the first two families of maximum scattered $\mathbb{F}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered $\mathbb{F}_q$-subspaces of $V(2,q^n)$ have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the $\mathbb{F}_q$-linear sets presented in [23] and in [5], for $n=6,8$, are new. Also, for $q$ odd, $q\equiv \pm 1,\,0 \pmod 5$, we present new examples of maximum scattered $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^6)$, ari…
We revisit and extend the connections between $\mathbb{F}_{q^m}$-linear rank-metric codes and evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We give a unifying framework in which we prove in an elementary way how the parameters of a rank-metric code are related to special geometric properties of the associated evasive subspace, with a particular focus on the generalized rank weights. In this way, we can also provide alternative and very short proofs of known results on scattered subspaces. We then use this simplified point of view in order to get a geometric characterization of near…
In this paper we investigate the geometric properties of the configuration consisting of a $k$-subspace $Γ$ and a canonical subgeometry $Σ$ in $\mathrm{PG}(n-1,q^n)$, with $Γ\capΣ=\emptyset$. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of $Σ$ from the vertex $Γ$. In particular we deal with the maximum scattered linear sets of the line $\mathrm{PG}(1,q^n)$ found by Lunardon and Polverino and recently generalized by Sheekey. Our aim is to characterize this family by means of the properties of the vertex of the project…
Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $\mathrm{PG}(1,q^t)$. In this paper a similar construction of a translation plane $\mathcal A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\mathbb F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $\mathbb F_q$-subspace of $\mathbb F_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $\mathcal A_f$ and $\mathcal A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f…
More than twenty-five years ago, Manickam, Miklós, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. We verify this conjecture when $n \geq 8k^{2}$, which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when $k < 10^{45}$. Moreover, our arguments resolve the vector space analogue of this conjecture. Let $V$ be an $n$-dimensional vector space over a finite field. Assign a real-value…
For a prime power $q$ and a positive integer $n$, we say a subspace $U$ of ${\mathbb{F}_q^n}$ is {\em cyclically covering} if the union of the cyclic shifts of $U$ is equal to $\mathbb{F}_q^n$. We investigate the problem of determining the minimum possible dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. (This is a natural generalisation of a problem posed in 1991 by the first author.) We prove several upper and lower bounds, and for each fixed $q$, we answer the question completely for infinitely many values of $n$ (which take the form of certain geometric series). Our results…
A subspace of $\mathbb{F}_2^n$ is called cyclically covering if every vector in $\mathbb{F}_2^n$ has a cyclic shift which is inside the subspace. Let $h_2(n)$ denote the largest possible codimension of a cyclically covering subspace of $\mathbb{F}_2^n$. We show that $h_2(p)= 2$ for every prime $p$ such that 2 is a primitive root modulo $p$, which, assuming Artin's conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds on $h_2(ab)$ depending on $h_2(a)$ and $h_2(b)$ and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud.
In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski's fundamental theorem, and which is strongly linked to the well-known Mahler-conjecture. Based on the covering minima of Kannan & Lovász and a problem posed by Fejes Tóth, we arrange Makai Jr.'s conjecture into a wider context and investigate densities of lattice arrange…
On the existence of linear rank-metric intersecting codes
Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA, Université Sorbonne Paris Nord, CNRS（巴黎第八大学，几何、分析与应用实验室，LAGA，索邦巴黎北大学，法国国家科学研究中心）；Inria Saclay（萨克雷英ria研究院）；Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”（坎帕尼亚“路易吉·万维塔利”大学数学与物理系）
The linear spaces that are fixed by a given nilpotent $n \times n$ matrix form a subvariety of the Grassmannian. We classify these varieties for small $n$. Mutiah, Weekes and Yacobi conjectured that their radical ideals are generated by certain linear forms known as shuffle equations. We prove this conjecture for $n \leq 7$, and we disprove it for $n=8$. The question remains open for nilpotent matrices arising from the affine Grassmannian.
In this paper we give new constructions of Ferrer diagram rank metric codes, which achieve the largest possible dimension. In particular, we prove several cases of a conjecture by T. Etzion and N. Silberstein. We also establish a sharp lower bound on the dimension of linear rank metric anticodes with a given profile. Combining our results with the multilevel construction, we produce examples of subspace codes with the largest known cardinality for the given parameters.
We give one more proof of the first linear programming bound for binary codes, following the line of work initiated by Friedman and Tillich. The new argument is somewhat similar to previous proofs, but we believe it to be both simpler and more intuitive. Moreover, it provides the following 'geometric' explanation for the bound. A binary code with minimal distance $δn$ is small because the projections of the characteristic functions of its elements on the subspace spanned by the Walsh-Fourier characters of weight up to $\left(\frac 12 - \sqrt{δ(1-δ)}\right) \cdot n$ are essentially independent.…
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations, Ministry of Education（教育部非线性预期前沿科学中心）
研究向量空间中 Erdős 匹配猜想的类比，通过构造和组合方法证明了对 k=2、n=(s+1)k 及 n 足够大时猜想成立，并得到稳定性定理和极值数结果。
We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.
Let $p$ be a prime, and let $\mathbb{Z}_p$ denote the field of integers modulo $p$. The \emph{Nathanson height} of a point $v \in \mathbb{Z}_p^n$ is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace $V \subseteq \mathbb{Z}_p^n$ is the least Nathanson height of any of its nonzero points. In this paper, we resolve a conjecture of Nathanson [M. B. Nathanson, Heights on the finite projective line, International Journal of Number Theory, to appear], showing that on subspaces of $\mathbb{Z}_p^n$ of codimension one, the Nathanson height fu…
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 $\mathcal{O}_n \subset \mathcal{P}_q$ the set of polynomials that are either the zero polynomial, or have at most $n$ distinct roots in $\mathbb{F}_q$. Given two subspaces $Y,Z$ of $\mathcal{P}_q$, we denote by $\langle Y,Z \rangle$ their span. We prove that the following are equivalent. A) Let $k, q$ integers, with $q$ a prime power and $2 \leq k \leq q$. Suppose that eith…
