Equidistribution for nilsequences along spheres over finite fields

A note on complex conference matrices

This paper introduces a construction principle for generating matrices of digital sequences over a finite field \(\mathbb{F }_q\), which is based on sequences of polynomials and their representations in terms of powers of nonconstant polynomials. For the most basic polynomial sequence, \((x^r)_{r\ge 0}\), the representations in terms of powers of linear polynomials yield, within this construction principle, the Pascal matrices, which consist of binomial coefficients and were earlier introduced by Faure for finite prime fields and by Niederreiter for finite field extensions. Generally, for binomial type sequences of polynomials an interesting relation between the generating matrices is worked out, and further examples of generating matrices are given, which contain combinatorial magnitudes as, e.g., binomial coefficients, Stirling numbers of the first kind, Stirling numbers of the second kind, and Bell numbers. Moreover, within this construction principle, explicit constructions of finite-row generating matrices of digital \((t,s)\)-sequences are presented, which were so far only known for \(t\) equal to \(0\).

Recently, Kim and Son [9] have obtained an infinite sequence of irreducible polynomials over the finite field Fq of characteristic p by using the transformation . Here, it is shown that one can also build this infinite sequence of irreducible polynomials over the field Fq of characteristic p by using a different transformation . Moreover, this transformation is helpful to establish the sequence of normal polynomials of degree npr in Fq [x] by using an appropriate initial normal polynomial of degree n.

Some results on the [Formula: see text]-normal elements and [Formula: see text]-normal polynomials over finite fields are given in the recent literature. In this paper, we show that a transformation [Formula: see text] can be used to produce an infinite sequence of irreducible polynomials over a finite field [Formula: see text] of characteristic [Formula: see text]. By iteration of this transformation, we construct the [Formula: see text]-normal polynomials of degree [Formula: see text] in [Formula: see text] starting from a suitable initial [Formula: see text]-normal polynomial of degree [Formula: see text]. We also construct an infinite sequence of [Formula: see text]-normal polynomials using a certain quadratic transformation over [Formula: see text].

On the fractal structure of the rescaled evolution set of Carlitz sequences of polynomials

Given ε > 0 \varepsilon > 0 , we show that there are infinitely many sequences of consecutive ε n \varepsilon n -smooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ⁡ ln ⁡ ln ⁡ n \ln \ln \ln n .

Let χ be a nontrivial multiplicative character of Fpn. We obtain the following results. (1) Let ϵ>0 be given. If B={∑j=1nxjωj :xj∈[Nj+1,Nj+Hj]∩Z,j=1,…,n} is a box satisfying Πj=1nHj>p(2/5+ϵ)n, then for p>p(ϵ) we have, denoting χ a nontrivial multiplicative character, |∑x∈Bχ(x)|≪np-ϵ2/4|B| unless n is even, χ is principal on a subfield F2 of size pn/2, and maxξ|B∩ξF2|>p-ϵ|B|. (2) Assume that A,B⊂Fp so that |A|>p(4/9)+ϵ, |B|>p(4/9)+ϵ, |B+B| 1/2+δ. Then for a nonprincipal multiplicative character χ, |∑x∈I,y∈Dχ(x+y)|<p-δ2/12|I| |D|. We also slightly improve a result of Karacuba [K3]

Let p be an odd rational prime, q = p s, s ≥ 1, and F q be the finite field of q elements. Let ε denote the trivial multiplicative character on F * q and η denote the quadratic character on F * q . The trivial character η is extended to all of F q by setting η(0) = 1. If ψ is a nontrivial character on F * q we define ψ(0) = 0.

On the Distribution of Powers in Finite Fields

For a nontrivial additive character λ of the finite field with q elements and each positive integer r, the exponential sums ∑ λ ( ( trw )r ) over w ∈ SO+(2n,q) and over w ∈ O+(2n,q) are considered. We show that both of them can be expressed as polynomials in q involving certain new exponential sums. Estimates on those new exponential sums are given. Also, we derive from these expressions the formulas for the number of elements w in SO+(2n,q) and O+(2n,q) with (trw) r = β, for each β in the finite field with q elements.

The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character chi, either chi(P) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of chi(P) and its "structure"?&#13;\n&#13;\nPreviously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field F_p. Our technical results are:&#13;\n&#13;\n1. If P: K^n -&gt; K is a polynomial of degree &lt;= d, and E[chi(P(x))] &gt; |K|^{-s} for some s &gt; 0 and non-trivial additive character chi, then P is a function of O_{d, s}(1) many non-classical polynomials of weight degree &lt; d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work.&#13;\n&#13;\n2. Suppose K and F are of bounded order, and let H be an affine subspace of K^n. Then, if P: K^n -&gt; K is a polynomial of degree d that is sufficiently regular, then (P(x): x in H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P. Such a theorem was previously known for H an affine subspace over a prime field.&#13;\n&#13;\n&#13;\nThe tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas.&#13;\n&#13;\n(i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code.&#13;\n&#13;\n(ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P: K^n -&gt; K can be decomposed as a particular composition of lesser degree polynomials.&#13;\n&#13;\n(iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f: K^n -&gt; K are testable with one-sided error.

For a nontrivial additive character \( \lambda \) of the finite field with q elements and each positive integer r, the 'Gauss' sums \( \sum \lambda ({(\rm tr}\, g)^r) \) over \( g \in {\rm GL}\,(n,q) \) and over \( g \in {\rm SL}\,(n,q) \) are considered. We show that both of them can be expressed as polynomials in q with coefficients involving certain exponential sums. As applications, we derive from these expressions the formulas for the number of elements g in GL (n,q) and SL (n,q) with \( {\rm tr}\, g = \beta \), for each \( \beta \) in the finite field with q elements.

This paper presents some results with the constructive theory of synthesis of irreducible polynomials over a Galois field with even characteristic. We prove a theorem that plays an important role for constructing irreducible polynomials. By this theorem two recurrent methods for constructing families of irreducible polynomials of degree \(n2^{k}~(k=1,2,\ldots )\) over \(\mathbb F _{2^{s}}\) are proposed. It is shown that in this special case, the sequences of irreducible polynomials are N-polynomial of degree \(2^{k}\).

A recently proposed public key cryptosystem based on Chebyshev polynomials suggests a new approach to data encryption. The sequence of Chebyshev polynomial is proved to be periodically, but its symmetry properties have not been investigated in depth. In this paper, a new representation of Chebyshev polynomial is introduced to study these properties, and their impacts on the cryptosystem. It is shown that the factual private key space of the cryptosystem is only half or quarter of the period of Chebyshev polynomial sequence. Proper parameters should be chosen to ensure the security of the cryptosystem.

Recursive constructions of irreducible polynomials over finite fields