Exponential sums for skew matrices in a finite field

1956 ◽  
Vol 7 (2) ◽  
pp. 116-121 ◽  
Author(s):  
John H. Hodges
Keyword(s):  
Finite Field ◽  
Exponential Sums

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

DEGREE MATRICES AND ESTIMATES FOR EXPONENTIAL SUMS OF POLYNOMIALS OVER FINITE FIELDS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350030 ◽  
Author(s):  
WEI CAO
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Exponential Sums ◽  
Elementary Approach ◽  
Multivariate Polynomial ◽  
Elementary Divisors ◽  
Polynomials Over Finite Fields ◽  
Degree Matrix

Let f be a multivariate polynomial over a finite field and its degree matrix be composed of the degree vectors appearing in f. In this paper, we provide an elementary approach to estimating the exponential sums of the polynomials with positive square degree matrices in terms of the elementary divisors of the degree matrices.

scholarly journals Infinite Families of 2-Designs from a Class of Linear Codes Related to Dembowski-Ostrom Functions

2021 ◽  
pp. 1-15
Author(s):  
Rong Wang ◽  
Xiaoni Du ◽  
Cuiling Fan ◽  
Zhihua Niu
Keyword(s):  
Finite Field ◽  
Weight Distribution ◽  
Coding Theory ◽  
Exponential Sums ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Research Interest ◽  
Combinatorial Formula ◽  
Text Design ◽  
Fixed Weight

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial [Formula: see text]-designs have attracted lots of research interest for decades. The interplay between coding theory and [Formula: see text]-designs started many years ago. It is generally known that [Formula: see text]-designs can be used to derive linear codes over any finite field, and that the supports of all codewords with a fixed weight in a code also may hold a [Formula: see text]-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then determine the weight distribution of the linear codes. Finally, we obtain infinite families of [Formula: see text]-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of [Formula: see text]-designs are calculated explicitly.

scholarly journals On the set of distances between two sets over finite fields

2006 ◽  
Vol 2006 ◽  
pp. 1-5 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Exponential Sums

We use bounds of exponential sums to derive new lower bounds on the number of distinct distances between all pairs of points(x,y)∈×ℬfor two given sets,ℬ∈Fqn, whereFqis a finite field ofqelements andn≥1is an integer.

EXPONENTIAL SUMS ON 𝔸n. IV

2009 ◽  
Vol 05 (05) ◽  
pp. 747-764 ◽  
Author(s):  
ALAN ADOLPHSON ◽  
STEVEN SPERBER
Keyword(s):  
Finite Field ◽  
Cohomology Group ◽  
Exponential Sums ◽  
Exponential Sum

We find new conditions on a polynomial over a finite field that guarantee that the exponential sum defined by the polynomial has only one nonvanishing p-adic cohomology group, hence the L-function associated to the exponential sum is a polynomial or the reciprocal of a polynomial.

Exponential Sums for Symmetric Matrices in a Finite Field

1955 ◽  
Vol 14 (4-6) ◽  
pp. 331-339 ◽  
Author(s):  
John H. Hodges
Keyword(s):  
Finite Field ◽  
Exponential Sums ◽  
Symmetric Matrices

Convolution and Equidistribution

2012 ◽  
Author(s):  
Nicholas M. Katz
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Mellin Transform ◽  
Exponential Sums ◽  
Multiplicative Group ◽  
Input Function ◽  
Point Of View ◽  
Basic Question ◽  
Mellin Transforms ◽  
The Subject

This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

A NOTE ON SOME CHARACTER SUMS OVER FINITE FIELDS

2015 ◽  
Vol 92 (1) ◽  
pp. 32-43
Author(s):  
XIWANG CAO ◽  
GUANGKUI XU
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Boolean Functions ◽  
Exponential Sums ◽  
Character Sums ◽  
Walsh Spectrum

In this paper, we present a decomposition of the elements of a finite field and illustrate the efficiency of this decomposition in evaluating some specific exponential sums over finite fields. The results can be employed in determining the Walsh spectrum of some Boolean functions.

scholarly journals ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE

2010 ◽  
Vol 52 (2) ◽  
pp. 315-324 ◽  
Author(s):  
VALÉRIE GILLOT ◽  
PHILIPPE LANGEVIN
Keyword(s):  
Finite Field ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Spectral Amplitude

AbstractIn this paper, we improve results of Gillot, Kumar and Moreno to estimate some exponential sums by means of q-degrees. The method consists in applying suitable elementary transformations to see an exponential sum over a finite field as an exponential sum over a product of subfields in order to apply Deligne bound. In particular, we obtain new results on the spectral amplitude of some monomials.

scholarly journals On some generalisations of the Erdős distance problem over finite fields

2006 ◽  
Vol 73 (2) ◽  
pp. 285-292 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Exponential Sums ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points (a, b) ∈ A × B for two given sets where is a finite field of q elements and n ≥ 1 is an integer.

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