Duality for digital sequences
type: Item
instance of: scholarly article
title: Duality for digital sequences
zbMATH Open document ID: 1181.11049
author: Josef Dick
author: Harald Niederreiter
published in: Journal of Complexity
publication date: 2009-10-09T00:00:00Z
review text: The \((t,m,s)-\)nets and the \((t,s)-\)sequences are classes of objects with an exceptionally well distribution in the \(s-\)dimensional unit cube \([0,1)^{s}.\) They have numerous applications in many branches of science. There are many analytical algorithms for their construction. The duality theory for digital nets has been introduced by \textit{H. Niederreiter} and \textit{G. Pirsic} [Acta Arith. 97, 173--182 (2001; Zbl 0972.11066)]. In this paper the duality theory is extended from finite digital nets to infinite digital sequences. In the Introduction the common concepts of the construction of digital nets and sequences over finite fields based on using generating matrices is reminded. The duality theory for digital nets is briefly presented. In Section 2 the definition of digital nets and sequences over finite fields with respect to an arbitrary bijection which is based on using generating matrices is reminded. The dual space \({\mathcal N}_{m}^{\bot}\) of the mth row space \({\mathcal N}_{m}\) of digital sequence is introduced. In Section 3 the duality theory for digital nets is extended to duality theory for digital sequences. The notion of minimum distance is defined. The necessary and sufficient conditions for generating a digital \(({\mathbf T},s)-\)sequences and a strict digital \(({\mathbf T},s)-\)sequences in the terms of minimum distance are obtained. In Section 4 the fundamental properties, that show which sequences of the dual space define a digital sequence and how the digital sequence can be obtained from the sequence \(({\mathcal M}_{m})_{m \geq 1}\) of the dual space chain, are obtained. In Definition 3 the notion of dual space chain is introduced. Theorem 1 confirms that for a given generating matrices \(C_{1}, \dots, C_{s}\) of digital sequences the associated sequence \(({\mathcal N}_{m}^{\bot})_{m \geq 1}\) is a dual space chain. Theorem 2 shows that for a given sequence \(({\mathcal M}_{m})_{m \geq 1}\) of dual space chain one can construct…
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65C10
Mathematics Subject Classification ID: 65C40
Mathematics Subject Classification ID: 11K31
zbMATH DE Number: 5611796
zbMATH Keywords: \((t, m, s)\)-nets and \((t, s)\)-sequences over finite fields
zbMATH Keywords: duality theory
zbMATH Keywords: dual space chain
zbMATH Keywords: sequences of Niederreiter and Xing
describes a project that uses: MinT
reviewed by: Vassil Grozdanov
MaRDI profile type: Publication
full work available at URL: j.jco.2009.06.004
OpenAlex ID: W2166008912
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DOI: 10.1016/J.JCO.2009.06.004
description: scientific article; zbMATH DE number 5611796