Finite Vector Space Research Articles

The classification of Boolean degree 1 functions in high-dimensional finite vector spaces

We classify the Boolean degree 1 1 functions of k k -spaces in a vector space of dimension n n (also known as Cameron-Liebler classes) over the field with q q elements for n ≥ n 0 ( k , q ) n \geq n_0(k, q) . This also implies that two-intersecting sets with respect to k k -spaces do not exist for n ≥ n 0 ( k , q ) n \geq n_0(k, q) . Our main ingredient is the Ramsey theory for geometric lattices.

Vector multispaces and multispace codes

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of Fqn) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q.

Nontrivial t-designs in polar spaces exist for all t

AbstractA finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over $$\mathbb {F}_q$$ F q equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-$$(n,k,\lambda )$$ ( n , k , λ ) design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly $$\lambda $$ λ members of Y. Nontrivial examples are currently only known for $$t\le 2$$ t ≤ 2 . We show that t-$$(n,k,\lambda )$$ ( n , k , λ ) designs in polar spaces exist for all t and q provided that $$k>\frac{21}{2}t$$ k > 21 2 t and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

Technical note: Determining chemical composition of atmospheric single particles by a standard-free mass calibration algorithm

Abstract. The chemical composition of individual particles can be revealed by single-particle mass spectrometers (SPMSs). With higher accuracy in the ratio of mass to charge (m/z), more detailed chemical information could be obtained. In SPMSs, the conventional standard-based calibration methods (internal/external) are constrained by the inhomogeneity of ionization lasers and the finite focusing ability of the inlet system, etc.; therefore, the mass accuracy is restricted. In this study, we obtained the detailed and reliable chemical composition of single particles utilizing a standard-free mass calibration algorithm. In the algorithm, the characteristic distributions of hundreds of ions were concluded and collected in a database denoted as prototype. Each single-particle mass spectrum was initially calibrated by a function with specific coefficients. The range of coefficients was constrained by the magnitude of mass deviation to a finite vector space. To find the optimal coefficient vector, the conformity of each initially calibrated spectrum to the prototype dataset was assessed. The optimum calibrated spectrum was obtained with maximum conformity. For more than 98 % ambient particles, a 20-fold improvement in mass accuracy, from ∼ 10 000 ppm (integer) to ∼ 500 ppm (two decimal places), was achieved. The improved mass accuracy validated the determination of adjacent ions with a m/z difference ∼ 0.05 Th. Furthermore, atmospheric particulate trace elements that were poorly studied before are specified. The obtained detailed single-particle-level chemical information could help explain the source apportionment, reaction mechanism, and mixing state of atmospheric particles.

Random walks and the “Euclidean” association scheme in finite vector spaces

Abstract In this paper, we provide an application to the random distance-t walk in finite planes and derive asymptotic formulas (as $q \to \infty $ ) for the probability of return to start point after $\ell $ steps based on the “vertical” equidistribution of Kloosterman sums established by N. Katz. This work relies on a “Euclidean” association scheme studied in prior work of W. M. Kwok, E. Bannai, O. Shimabukuro, and H. Tanaka. We also provide a self-contained computation of the P-matrix and intersection numbers of this scheme for convenience in our application as well as a more explicit form for the intersection numbers in the planar case.

Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces

Abstract The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in 𝔽 q 2 {\mathbb{F}_{q}^{2}} . More precisely, for A , B , C ⊂ 𝔽 q 2 {A,B,C\subset\mathbb{F}_{q}^{2}} , if | A | ⁢ | B | ⁢ | C | 1 2 ≫ q 4 {|A||B||C|^{\frac{1}{2}}\gg q^{4}} , then for any λ ∈ 𝔽 q ∖ { 0 } {\lambda\in\mathbb{F}_{q}\setminus\{0\}} , the number of congruence classes of triangles with vertices in A × B × C {A\times B\times C} and one side-length λ is at least ≫ q 2 {\gg q^{2}} . In higher dimensions, we obtain similar results for k-simplex but under a slightly stronger condition. Compared to the well-known L 2 {L^{2}} method in the literature, our approach offers better results in both conditions and conclusions. When A = B = C {A=B=C} , the second goal of this paper is to give a new and unified proof of the best current results on the distribution of simplex due to Bennett, Hart, Iosevich, Pakianathan and Rudnev (2017) and McDonald (2020). The third goal of this paper is to study a Furstenberg-type problem associated to a set of rigid motions. The main ingredients in our proofs are incidence bounds between points and rigid motions. While the incidence bounds for large sets are due to the author and Semin Yoo (2023), the bound for small sets will be proved by using a point–line incidence bound in 𝔽 q 3 {\mathbb{F}_{q}^{3}} due to Kollár (2015).

Application of Clifford Algebra on Group Theory

The orthogonal operators defined as similarity transformations on Euclidean space E can also be considered as group actions on the Clifford Algebra. In this paper, we investigate the finite subgroup of Euclidian space E of Geometric Algebra over a finite dimension vector space E. The hierarchy of the finite subgroups of Clifford Algebra C(E) is depicted through the lattice structure and we discussed the group action of these subgroups on the vector space E. Further, we shall address the number of non-trivial finite subgroups, Normal subgroups, and subnormal series of the subgroup of Clifford Algebra C(E) constructed over the vector space E by performing group action over the subgroup of Clifford Algebra C(E).

Almost spanning distance trees in subsets of finite vector spaces

AbstractFor and an odd prime power , consider the vector space over the finite field , where the distance between two points and is defined as . A distance graph is a graph associated with a nonzero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every nearly spanning distance tree with bounded degree in each distance. This quantitatively improves results by Bennett, Chapman, Covert, Hart, Iosevich, and Pakianathan on finding distance paths, and results by Pham, Senger, Tait, and Thu on finding distance trees. A key ingredient in proving our main result is to obtain a colorful generalization of a classical result of Haxell about finding nearly spanning bounded‐degree trees in an expander.

Enumeration of anti-invariant subspaces and Touchard's formula for the entries of the q-Hermite Catalan matrix

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

On the G-quadratic and LG-quadratic of the Exterior Algebra and Associated Algebra

Given an exterior algebra over a finite dimension vector space v, and let , where is a graded ideal in . The relation between the algebra and regarding to -quadratic and - quadratic will be investigated. We show that the algebra is - quadratic if and only if is - quadratic. Furthermore, it has been shown that the algebra is - quadratic if and only if is - quadratic.

Projective tilings and full-rank perfect codes

A tiling of a vector space S is the pair (U, V) of its subsets such that every vector in S is uniquely represented as the sum of a vector from U and a vector from V. A tiling is connected to a perfect codes if one of the sets, say U, is projective, i.e., the union of one-dimensional subspaces of S. A tiling (U, V) is full-rank if the affine span of each of U, V is S. For finite non-binary vector spaces of dimension at least 6 (at least 10), we construct full-rank tilings (U, V) with projective U (both U and V, respectively). In particular, that construction gives a full-rank ternary 1-perfect code of length 13, solving a known problem. We also discuss the treatment of tilings with projective components as factorizations of projective spaces.

The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers $$e_1,e_2$$ , let $$X_i$$ denote the set of $$e_i$$ -dimensional subspaces of a fixed finite vector space $$V=({\mathbb F}_q)^{e_1+e_2}$$ . Let $$Y_i$$ be a non-empty subset of $$X_i$$ and let $$\alpha _i = |Y_i|/|X_i|$$ . We give a positive lower bound, depending only on $$\alpha _1,\alpha _2,e_1,e_2,q$$ , for the proportion of pairs $$(S_1,S_2)\in Y_1\times Y_2$$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.

A Kruskal–Katona-type theorem for graphs: q-Kneser graphs

The “Kruskal-Katona-type problem for a graph G” concerned here is to describe subsets of vertices of G that have minimum number of neighborhoods with respect to their sizes. In this paper, we establish a Kruskal-Katona-type theorem for the q-Kneser graphs, whose vertex set consists of all k-dimensional subspaces of an n-dimensional linear space over a q-element field, two subspaces are adjacent if they have the trivial intersection. It includes as a special case the Erdős–Ko–Rado theorem for intersecting families in finite vector spaces and yields a short proof of the Hilton-Milner theorem for nontrivial intersecting families in finite vector spaces.

Subset sums and block designs in a finite vector space

In this paper we settle the question of whether a finite-dimensional vector space {{mathcal {V}}} over {mathbb {F}}_p, with p an odd prime, and the family of all the k-sets of elements of {mathcal {V}} summing up to a given element x, form a 1-(v,k,lambda _1) or a 2-(v,k,lambda _2) block design, and, in either case, we find a closed form for lambda _i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The “twin case” p=2, which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author.

Packings and Steiner systems in polar spaces

A finite classical polar space of rank \(n\) consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that \(n\) is the maximal dimension of such a subspace. A \(t\)-Steiner system in a finite classical polar space of rank \(n\) is a collection \(Y\) of totally isotropic \(n\)-spaces such that each totally isotropic \(t\)-space is contained in exactly one member of \(Y\). Nontrivial examples are known only for \(t=1\) and \(t=n-1\). We give an almost complete classification of such \(t\)-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.Mathematics Subject Classifications: 51E23, 05E30, 33C80Keywords: Association schemes, codes, polar spaces, Steiner systems

The covering lemma and q-analogues of extremal set theory problems

We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.

Threshold functions for incidence properties in finite vector spaces

The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. Specifically, we consider the events that there is an ℓ-rich m-flat with regard to a random set of points in Fqn, the event that a random set of points is an m-blocking set, and the event that there is an incidence between a random set of points and a random set of m-flats. One of our key ingredients is a stronger version of a recent result obtained by Chen and Greenhill (2021).

Taming numerical errors in simulations of continuous variable non-Gaussian state preparation

Numerical simulation of continuous variable quantum state preparation is a necessary tool for optimization of existing quantum information processing protocols. A powerful instrument for such simulation is the numerical computation in the Fock state representation. It unavoidably uses an approximation of the infinite-dimensional Fock space by finite complex vector spaces implementable with classical digital computers. In this approximation we analyze the accuracy of several currently available methods for computation of the truncated coherent displacement operator. To overcome their limitations we propose an alternative with improved accuracy based on the standard matrix exponential. We then employ the method in analysis of non-Gaussian state preparation scheme based on coherent displacement of a two mode squeezed vacuum with subsequent photon counting measurement. We compare different detection mechanisms, including avalanche photodiodes, their cascades, and photon number resolving detectors in the context of engineering non-linearly squeezed cubic states and construction of qubit-like superpositions between vacuum and single photon states.

Regular orbits of finite primitive solvable groups, the final classification

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V, and G is not metacyclic. Then G always has a regular orbit on V except for a few “small” cases. We completely classify these cases in this paper.

Probabilistic relations for transformations of finite vector spaces

Рассматриваются отображения конечномерных векторных пространств над конечным полем, которые задаются функциональной схемой, состоящей из линейной среды и локальных нелинейных преобразований. Доказаны теоремы о выражении точных значений вероятностей выполнения линейных и разностных соотношений через аналогичные характеристики для локальных преобразований.