A transform property of Kloosterman sums
type: Item
instance of: scholarly article
title: A transform property of Kloosterman sums
zbMATH Open document ID: 1268.11107
author: Ian Blake
author: Theo Garefalakis
published in: Discrete Applied Mathematics
publication date: 2010-10-11T00:00:00Z
review text: Kloosterman sums \(K_k(a,b)\) are of interest in many parts of mathematics. They are defined as \(K_k(a,b)=\sum_{\gamma \in {\mathbb{F}_{q^k}^\ast}}\chi(\text{trace}(a\gamma+b\gamma^{-1})\), where \(\chi\) in an additive character of the finite field \({\mathbb F}_{q}\). Here \(q\) is a prime power and the trace is relative to \({\mathbb F}_q\). These numbers are related to the number \(n_k(\beta, a, b)\) of solutions \(\gamma\) such that \(\text{trace}(a\gamma+b\gamma^{-1})=\beta\). Some interesting recursion formulae for the \(n_k\) are observed, and these are used to count the number of certain irreducible polynomials as well as simplifying proof of a result on the dual weight distribution of a Melas code by \textit{J. C. C. M. Remin} and \textit{H. J. Tiersma} [IEEE Trans. Inf. Theory 34, No. 5, Part II, 1348--1351 (1989; Zbl 0666.94013)].
reviewed by: Alexander Pott
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 5798498
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: character sum
zbMATH Keywords: finite field
zbMATH Keywords: Melas code
MaRDI profile type: Publication
full work available at URL: j.dam.2010.02.012
OpenAlex ID: W2036081567
cites work: Q4265362
cites work: Kloosterman sums and finite field extensions
cites work: The divisibility modulo 24 of Kloosterman sums on \(\text{GF}(2^m)\), \(m\) odd
cites work: Explicit theorems on generator polynomials
cites work: On a new identity for Kloosterman sums and nonlinear system of equations over finite fields of characteristic 2.
cites work: Q4706961
cites work: Exponential sums and coding theory: A review
cites work: Q3478531
cites work: Distribution of the weights of the dual of the Melas code
cites work: Q3797302
cites work: Q4146667
cites work: The divisibility modulo 24 of Kloosterman sums on \(\text{GF}(2^m)\), \(m\) even
cites work: Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm
cites work: A duality theorem for the weight distribution of some cyclic codes
cites work: Families of curves and weight distributions of codes
cites work: Hecke operators and the weight distributions of certain codes
cites work: A new Kloosterman sum identity over \(F_{2^m}\) for odd \(m\).
cites work: The weights of the dual code of the MELAS code over GF(3)
cites work: Irreducible polynomials over GF(2) with prescribed coefficients.
DOI: 10.1016/J.DAM.2010.02.012
description: scientific article; zbMATH DE number 5798498