The Rado multiplicity problem in vector spaces over finite fields

We study the L[superscript p] - L[superscript r] boundedness of the extension operator associated with algebraic varieties such as nondegenerate quadratic surfaces, paraboloids, and cones in vector spaces over finite fields. We obtain the best possible result for the extension theorems related to nondegenerate quadratic curves in two dimensional vector spaces over finite fields. In higher even dimensions, we improve upon the Tomas-Stein exponents which were obtained by Mockenhaupt and Tao ([21]) by studying extension theorems for paraboloids in the finite field setting. We also study extension theorems for cones in vector spaces over finite fields. We give an alternative proof of the best possible result for the extension theorems for cones in three dimensions, which originally is due to Mockenhaupt and Tao ([21]). Moreover, our method enables us to obtain the sharp L² - L[superscript r] estimate of the extension operator for cones in higher dimensions. In addition, we study the relation between extension theorems for spheres and the Erdos-Falconer distance problems in the finite field setting. Using the sharp extension theorem for circles, we improve upon the best known result, due to A. Iosevich and M. Rudnev ([17]), for the Erdos-Falconer distance problems in two dimensional vector spaces over finite fields. Discrete Fourier analytic machinery, arithmetic considerations, and classical exponential sums play an important role in the proofs.

Prototypes of many combinatorial designs come from finite projective geometries and finite affine geometries . Vector spaces over finite fields provide a natural setting for describing these geometries. Among the numerous incidence structures that can be constructed using affine and projective geometries are infinite families of symmetric designs, nets and Latin squares . Subspaces of a vector space over a finite field can be regarded as linear codes that will be used in later chapters for constructing other combinatorial structures, such as Witt designs and balanced generalized weighing matrices . Finite fields In this section we recall a few basic results on finite fields which will be used throughout this book. For any prime p , the residue classes modulo p with the usual addition and multiplication form a finite field GF ( p ) of order p . These fields are called prime fields . Any finite field F of characteristic p contains GF ( p ) as a subfield. The field F then can be regarded as a finite-dimensional vector space over GF ( p ), and therefore, | F | = p n where n is the dimension of this vector space. Conversely, for any prime power q = p n , there is a unique (up to isomorphism) finite field of order q . This field is denoted by GF ( q ) and is often called the Galois field of order q . In general, the field GF ( q ) is isomorphic to (a unique) subfield of the field GF ( r ) if and only if r is a power of q .

We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q {\mathbb F}_q , the finite field with q q elements, by A ⋅ A + ⋯ + A ⋅ A A \cdot A+\dots +A \cdot A , where A A is a subset F q {\mathbb F}_q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdős-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdős-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdős-Falconer distance problem for subsets of the unit sphere in F q d \mathbb F_q^d and discuss their sharpness. This results in a reasonably complete description of the Erdős-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

Paley-like graphs over finite fields from vector spaces

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well-suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C subset Sigma^{K^n} is an r-query locally correctable code (LCC), where K is a finite field and Sigma is a finite alphabet, then the number of codewords in C is at most exp(O_{K, r, |Sigma|}(n^{r-1})). Also, we show that if C subset Sigma^{K^n} is an r-query locally testable code (LTC), then the number of codewords in C is at most \exp(O_{K, r, |Sigma|}(n^{r-2})). The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty and Sudan (ITCS 2013) construct affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM 2011) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, upto a small error in the Gowers norm.

We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas–Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role.

I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution to Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of linear independence among vectors; given a finite collection of vectors in a vector space, each subset is either dependent or independent. A matroid consists of a finite ground set together with a collection of subsets that we call independent; the independent sets satisfy natural combinatorial axioms coming from linear algebra. Not all matroids can be represented by a collection of vectors and, ever since their introduction by Hassler Whitney [26] in 1935, mathematicians have sought ways to characterize those matroids that are. Rota’s Conjecture asserts that representability over any given finite field is characterized by a finite list of obstructions. We will formalize these notions, and the conjecture, in the next section. In the remainder of this introduction, we will describe the journey that led us to a solution. In the late 1990s, Rota’s Conjecture was already known to hold for fields of size two, three, and

Generalized Galois numbers count the number of flags in vector spaces over finite fields. Asymptotically, as the dimension of the vector space becomes large, we give their exponential growth and determine their initial values. The initial values are expressed analytically in terms of theta functions and Euler’s generating function for the partition numbers. Our asymptotic enumeration method is based on a Demazure module limit construction for integrable highest weight representations of affine Kac-Moody algebras. For the classical Galois numbers that count the number of subspaces in vector spaces over finite fields, the theta functions are Jacobi theta functions. We apply our findings to the asymptotic number of linearqq-ary codes and conclude with some final remarks about possible future research concerning asymptotic enumerations via limit constructions for affine Kac-Moody algebras and modularity of characters of integrable highest weight representations.

We study the Erdös–Falconer distance problem in vector spaces over finite fields with respect to the cubic metric. Estimates for discrete Airy sums and Adolphson–Sperber estimates for exponential sums in terms of Newton polyhedra play a crucial role. Similar techniques are used to study the incidence problem between points and cubic and quadratic curves. As a result we obtain a nontrivial range of exponents that appear to be difficult to attain using combinatorial methods.

As an emerging sampling technique, Compressed Sensing provides a quite masterly approach to data acquisition. Compared with the traditional method, how to conquer the Shannon/Nyquist sampling theorem has been fundamentally resolved. In this paper, first, we provide deterministic constructions of sensing matrices based on vector spaces over finite fields. Second, we analyze two kinds of attributes of sensing matrices. One is the recovery performance with respect to compressing and recovering signals in terms of restricted isometry property. In particular, we obtain a series of binary sensing matrices with sparsity level that are quite better than some existing ones. In order to save the storage space and accelerate the recovery process of signals, another character sparsity of matrices has been taken into account. Third, we merge our binary matrices with some matrices owning low coherence in terms of an embedding manipulation to obtain the improved matrices still having low coherence. Finally, compared with the quintessential binary matrices, the improved matrices possess better character of compressing and recovering signals. The favorable performance of our binary and improved matrices have been demonstrated by numerical simulations.

On [formula omitted]-fold partitions of finite vector spaces and duality

In this paper, we construct a quantization functor, associating a complex vector space $\cal{H}(V)$ to a finite-dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp$(V )$. The main new technical result is a proof of a stronger form of the Stone–von Neumann property for the Heisenberg group $H(V )$. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group over finite field.

The subspace inclusion graph on a vector space [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set consists of nontrivial proper subspaces of [Formula: see text] and two vertices are adjacent if one is properly contained in another. In a recent paper, Das posed the following four conjectures on the subspace inclusion graph [Formula: see text]: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular. (3) [Formula: see text] is Hamiltonian. (4) [Formula: see text] is a Cayley graph. In the present paper, we prove the first two conjectures: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular.

We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional projections via projections of random sets. In the end we study the projections of sets which have the (discrete) Fourier decay.