An upper bound for the extended Kloosterman sums over Galois rings
type: Item
instance of: scholarly article
title: An upper bound for the extended Kloosterman sums over Galois rings
zbMATH Open document ID: 0917.11037
author: Abhijit G. Shanbhag
author: P. Vijay Kumar
author: Tor Helleseth
published in: Finite Fields and their Applications
publication date: 1999-08-02T00:00:00Z
review text: For a fixed prime \(p\) and integers \(e\geq 2\), \(m\geq 1\), write \(R_{e,m}\) for the Galois ring of characteristic \(p^e\) and containing \(p^{em}\) elements. Let \(\psi_{e,m}:R_{e,m}\to{\mathcal C}\) be an additive character of \(R_{e,m}\) and let \(f(x)\) be a nondegenerate polynomial over \(R_{e,m}\) with weighted degree \(D_{e,f}\). In 1995, Kumar and others obtained an expression for sums of the form \(\Sigma\psi_{a,m} (f(x))\), where the summation runs over the set of so-called Teichmüller representatives \({\mathcal T}_{e,m}\), leading to a certain bound for the absolute value of the sum. In the present paper, the authors obtain an upper bound for the so-called extended Kloosterman sums over Galois rings, i.e. exponential sums of the form \(\Sigma \psi_{e, m}(f_1(n)+ f_2(n^{-1}))\), \(f_1,f_2\in R_{e,m} [x]\), the summation running over \({\mathcal T}^*_{e,m}\), the non-zero elements of \({\mathcal T}_{e,m}\) referred to above. Some applications to sequence designs are also given.
reviewed by: A. R. Rajwade
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11T23
zbMATH DE Number: 1236197
zbMATH Keywords: Galois ring
zbMATH Keywords: upper bound
zbMATH Keywords: extended Kloosterman sums
zbMATH Keywords: exponential sums
MaRDI profile type: Publication
full work available at URL: ffta.1998.0211
OpenAlex ID: W2060440857
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DOI: 10.1006/FFTA.1998.0211
description: scientific article; zbMATH DE number 1236197