Linear recurring sequences over rings and modules
type: Item
instance of: scholarly article
title: Linear recurring sequences over rings and modules
zbMATH Open document ID: 0859.11001
DOI: 10.1007/BF02362772
author: V. L. Kurakin
author: Aleksey Kuzmin
author: A. V. Mikhalev
author: Alexander Nechaev
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-04-09T00:00:00Z
review text: The monograph under review deals with the algebraic theory of linear recurring sequences over rings and modules. This general approach makes it possible to give a foundation for the theory of multidimensional linear recurrences and to study their properties. By applying the powerful framework of commutative algebra many classical results about one-dimensional recurrences are generalized to the multidimensional case. On the other hand commutative algebra makes a benefit to the new theory, too. Interesting consequences of the theory of recurrences in ring theory are shown, in particular, in the theory of quasi Frobenius modules and in Hopf algebras. The authors pay special attention to periodic sequences and to properties of linear recurring sequences over finite fields. Applications to coding theory are studied, too. The monograph is divided into six chapters and more or less independently into 31 sections. Chapter 1, ``General properties of linear recurring sequences over modules'', introduces basic definitions of one- and multidimensional recurrence sequences; of periodic sequences, ideals and polynomials; of cycle type of linear recurrence sequences; of decimation of sequences. Central roles are played by the concepts of linear recurrence sequence families, annihilator ideals and systems of generators of linear recurrence sequences and families of linear recurrence sequences. In section 4 it is pointed out that it is possible to construct a quasi Frobenius module over any Artinian commutative ring over a linear recurrence sequence family of some principal ideal ring. The main object of Chapter 2, ``Linear recurring sequences over finite fields'', is to find generating systems of annihilator ideals of linear recurrence sequence families. The Berlekamp-Massey algorithm and Hopf algebras play a central role here. In Chapter 3, ``Linear recurring sequences over Artinian and finite rings'', the classical method of localization is used to characterize canonical generato…
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11T99
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 889748
zbMATH Keywords: applications to coding theory
zbMATH Keywords: statistical property
zbMATH Keywords: linear recurring sequences
zbMATH Keywords: multidimensional linear recurrences
zbMATH Keywords: quasi Frobenius modules
zbMATH Keywords: Hopf algebras
zbMATH Keywords: periodic sequences
zbMATH Keywords: finite fields
zbMATH Keywords: annihilator ideals
zbMATH Keywords: Berlekamp-Massey algorithm
zbMATH Keywords: Groebner bases
zbMATH Keywords: cycle types
zbMATH Keywords: coordinate sequences
zbMATH Keywords: maximal period length
reviewed by: Attila Pethő
MaRDI profile type: Publication
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Linear recurring sequences over rings and modules (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf02362772

Linear recurring sequences over rings and modules (OpenAlex ID)
OpenAlex ID: W2057616013