Moments, Exponential Sums, and Monodromy Groups

Rigid local systems with monodromy group the Conway group [formula omitted

In the analytic study of trace functions of ℓ -adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such transforms is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project.

We consider a large family of dynamical Belyi maps of arbitrary degree and study the arithmetic monodromy groups attached to the iterates of such maps. Building on the results of Bouw-Ejder-Karemaker on the geometric monodromy groups of these maps, we show that the quotient of the arithmetic monodromy group by the geometric monodromy group has order either 1 1 or 2 2 . Prior to this article, a result of this kind was only known for quadratic maps (Pink) and a few examples in degree 3 3 .

Subspace code is very useful in the error correction for random network coding. In this paper, subspace code based on flats in affine space over finite fields is constructed. Sphere-packing bound, Wang–Xing–Safavi-Naini bound, anticode bound, Ahlswede–Aydinian bound and Gilbert–Varshamov bound for the size of code based on flats in affine space over finite fields are provided. Finally, we introduce another method of obtaining the anticode bound for the size of code based on flats in affine space from Erdős–Ko–Rado theorem.

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient .

A simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field. For a polynomial f ( x , y ) f(x,y) over F q {F_q} of total degree n , our algorithm takes at most \[ n 4.89 log 2 n log ⁡ q {n^{4.89}}{\log ^2}n\log q \] operations in F q {F_q} to factor f ( x , y ) f(x,y) completely. This improves a probabilistic factorization algorithm of von zur Gathen and Kaltofen, which takes \[ O ( n 11 log ⁡ n log ⁡ q ) O({n^{11}}\log n\log q) \] operations to factor f ( x , y ) f(x,y) completely over F q {F_q} . The algorithm can be easily generalized to factor multivariate polynomials over finite fields. We shall give two further applications of the idea involved in the algorithm. One is concerned with exponential sums; the other is related to permutational polynomials over finite fields (a conjecture of Chowla and Zassenhaus).

By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

We defined the quadratic Hermitian forms on a space of even dimension over a finite field; they are constructed from classical Hermitian forms. We give the exact value of exponential sums associated to the quadratic Hermitian forms. On the other hand, we define with the space of quadratic Hermitian forms a new family of subcodes of the Reed-Muller code of order two, and we give the value of the parameters of these codes and of the associated trace codes.

Several classes of exponential sums and three-valued Walsh spectrums over finite fields

In [4], Carlitz determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz’s results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields. ∗franciscastr@gmail.com, Tel: (787)764-0000, X4696, Fax: (787)281-0651 †iverubio@uprrp.edu, Tel: (787)764-0000 (4759), Fax: (787)773-1717 ‡jmanuel jmvg@yahoo.com, Tel: 787-738-2161

Inversive methods provide an attractive alternative to linear methods for pseudorandom number and vector generation. We present the first nontrivial results on the distribution of sequences of digital inversive pseudorandom numbers and inversive pseudorandom vectors in parts of the period. The proofs are based on a new bound for certain exponential sums over finite fields.

On the L-functions associated with certain exponential sums

Curves with Many Points and Configurations of Hyperplanes over Finite Fields

In this paper, we present two new constructions of complex codebooks with multiplicative characters, additive characters and trace functions over finite fields, and determine the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, in the first construction, we generalize the result in [28]. In the second construction, we generalize the results in [12], we can achieve Welch bound for any odd prime p, we also derive the whole distribution of their inner products. The parameters of these codebooks are new.

Let k and K be commutative fields with dimkK=2n, n≥1 and char(k)≠2, 3. If k satisfies one of the following conditions: (1) k is a finite field and k contains a 3-th root of unity, or (2) K is a cyclic extension of k where k is not 3-closed, and k contains a 2a3b-th root of unity, where 6n=2a3bc and c is coprime to 2, 3, then there exists an embedding of the tn-dimensional affine space AG(tn,k) into the t-dimensional affine space AG(t,K), for all t≥2.