Ternary codes associated with \(\mathrm{O}(3,3^r)\) and power moments of Kloosterman sums with trace nonzero square arguments
type: Item
instance of: scholarly article
zbMATH Open document ID: 1280.11079
publication date: 2012-09-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.1156
full work available at URL: http://hrcak.srce.hr/index.php
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6083588
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: power moment
zbMATH Keywords: orthogonal group
zbMATH Keywords: weight distribution
author: Dae San Kim
MaRDI profile type: Publication
arXiv ID: 0909.1156
arXiv classification: math.NT
arXiv classification: cs.IT
arXiv classification: math.IT
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title: Ternary codes associated with \(\mathrm{O}(3,3^r)\) and power moments of Kloosterman sums with trace nonzero square arguments
published in: Mathematical Communications
review text: Let \(\lambda\) be the canonical additive character on a finite field \(\mathbb F_q\). The Kloosterman sum, for \(a\in \mathbb F_q^*\), is: NEWLINE\[NEWLINE K(a)=\sum_{\alpha\in\mathbb F_q^*} \lambda (\alpha +a\alpha^{-1}). NEWLINE\]NEWLINE For a positive integer \(h\), the \(h\)-th moment of the Kloosterman sum with trace nonzero square arguments is: NEWLINE\[NEWLINE T_{12}SK^h =\sum_{a\in \mathbb F_q^*, tr a\neq 0} K(a^2)^h. NEWLINE\]NEWLINE Now assume that \(q=3^r\). The author gives recursive formulas for \(T_{12}SK^h\). The formulas involve the Stirling numbers of the second kind and the weights of two ternary linear codes, one constructed from \(\mathrm{SO}(3,q)\) and the other from \(\mathrm{O}(3,q)\).
description: scientific article; zbMATH DE number 6083588