The eigenstructure of finite field trigonometric transforms
type: Item
instance of: scholarly article
title: The eigenstructure of finite field trigonometric transforms
zbMATH Open document ID: 1225.15011
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Linear Algebra and its Applications
publication date: 2011-07-15T00:00:00Z
review text: Let \(\mathrm{GF}(q)\) denote a finite field with \(q\) elements. Consider the set \(\mathrm{GI}(p)=\{a+jb\,|\,a,b\in\mathrm{GF}(p)\}\), where \(p\) is a prime such that \(j^2\equiv -1\,(\mathrm{mod}\,p)\) is a quadratic nonresidue over~\(\mathrm{GF}(p)\), i.e., \(p\equiv 3\,(\mathrm{mod}\,4)\). Let \(\zeta=a+jb\in\mathrm{GI}\,(p)\) satisfy \(a^2+b^2\equiv 1\,(\mathrm{mod}\,p)\), and let \(\mathrm{ord}\,(\zeta)\) denote its multiplicative order. Originating from \textit{R.~M.~Campello de Souza, H.~M.~de Oliveira, A.~N.~Kauffman} and \textit{A.~J.~A.~Paschoal} [Proc. IEEE Internat. Sympos. Informat. Theory, p.~293 (1998)], define \[ \cos_\zeta(x)=\frac{\zeta^x+\zeta^{-x}}{2},\quad \sin_\zeta(x)=\frac{\zeta^x-\zeta^{-x}}{2j},\quad x= 0,1,\dots,\mathrm{ord}\,(\zeta)-1. \] The authors present various types of trigonometric transformations that change a vector with entries in~\(\mathrm{GF}(p)\) into one with entries in~\(\mathrm{GI}(p)\). They find the eigenvalues of the transformation matrices and study their multiplicities. They also propose procedures for constructing corresponding eigenvectors. Finally, the authors suggest applications to multiuser communication systems and error-correcting codes.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 94A05
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5924562
zbMATH Keywords: eigenvalues
zbMATH Keywords: eigenvectors
zbMATH Keywords: finite fields
zbMATH Keywords: trigonometric transforms
zbMATH Keywords: multiuser communication systems
zbMATH Keywords: error-correcting codes
reviewed by: Jorma K. Merikoski
MaRDI profile type: Publication
full work available at URL: j.laa.2011.03.031
OpenAlex ID: W1966477666
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DOI: 10.1016/J.LAA.2011.03.031
description: scientific article; zbMATH DE number 5924562