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Linear Codes and Polylinear Recurrences over Finite Rings and Modules (A Survey)

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1719))

Abstract

We give a short survey of the results obtained in the last several decades that develop the theory of linear codes and polylinear recurrences over finite rings and modules following the well-known results on codes and polylinear recurrences over finite fields. The first direction contains the general results of theory of linear codes, including: the concepts of a reciprocal code and the MacWilliams identity; comparison of linear code properties over fields and over modules; study of weight functions on finite modules, that generalize in some natural way the Hamming weight on a finite field; the ways of representation of codes over fields by linear codes over modules. The second one develops the general theory of polylinear recurrences; describes the algebraic relations between the families of linear recurrent sequences and their periodic properties; studies the ways of obtaining “good” pseudorandom sequences from them. The interaction of these two directions leads to the results on the representation of linear codes by polylinear recurrences and to the constructions of recursive MDS-codes. The common algebraic foundation for the effective development of both directions is the Morita duality theory based on the concept of a quasi-Frobenius module.

The authors thank the referee for many interesting and helpful comments and suggestions that had improved the explication of the results.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics and Center of New Information Technologies, Moscow State University, Russia

    V. L. Kurakin, A. S. Kuzmin, V. T. Markov, A. V. Mikhalev & A. A. Nechaev

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  1. V. L. Kurakin
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  2. A. S. Kuzmin
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  3. V. T. Markov
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  4. A. V. Mikhalev
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  5. A. A. Nechaev
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Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, 2540 Dole St., # 483, Honolulu, HI, 96822, USA

    Marc Fossorier  & Shu Lin  & 

  2. Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Minato-ku, Tokyo, 106, Japan

    Hideki Imai

  3. AAECC/IRIT, Université Paul Sabatier, 118, Route de Narbonne, F31067, Toulouse, Cedex, France

    Alain Poli

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Kurakin, V.L., Kuzmin, A.S., Markov, V.T., Mikhalev, A.V., Nechaev, A.A. (1999). Linear Codes and Polylinear Recurrences over Finite Rings and Modules (A Survey). In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_36

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