Skip to main content
SpringerLink
Log in

Polylinear Recurring Sequences over a Bimodule

  • Conference paper
Formal Power Series and Algebraic Combinatorics

Abstract

The main goal of the paper is to extend some of the known results of the theory of polylinear recurring sequences over fields and their generalizations for sequences over modules with commutative rings of coefficients to the case of noncommutative rings of coefficients. Possible noncommutativity of the main ring causes to consider polylinear sequences over a bimodule. To estimate linear complexity of the sequences in question we consider the notion of polylinear (k-linear) shift register. In fact, the theory of polylinear recurring sequences over fields admits a rather complete extension in this generality if the main bimodule is an Artinian duality context.

The work is partially supported by the RFFR grants 99-01-00941 and 99-01-00382.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Gill, A.: Linear Sequential Circuits. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  2. Faith, C.: Algebra II. Ring Theory. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  3. Lidl, R. and Niederreiter, H.: Finite fields. Addison-Wesley, London (1983)

    MATH  Google Scholar 

  4. Carmichael, R. D.: On sequences of integers defined by recurrence relations. Quart. J. Pure Appl. Math. 48 (1920) 343–372

    Google Scholar 

  5. Ward, M.: The arithmetical theory of linear recurring series. Trans. Amer. Math. Soc. 35 (1933), No. 3, 600–628

    Article  MathSciNet  Google Scholar 

  6. Zierler, N.: Linear recurring sequences. J. Soc. Ind. Appl. Math. 7 (1959), No. 1, 31–48

    Article  MathSciNet  MATH  Google Scholar 

  7. Azumaya, G.: A duality theory for injective modules (Theory of quasi-Frobenius modules). Amer. J. Math. 81 (1959), No. 1, 249–278

    Article  MathSciNet  MATH  Google Scholar 

  8. Cerlienco, L., Mignotte, M., and Piras, F.: Linear recurrent sequences: algebraic and arithmetical properties. Enseign. Math. (2) 33 (1987), No. 1–2, 67–108

    MathSciNet  MATH  Google Scholar 

  9. Kurakin, V. L., Kuzmin, A. S., Mikhalev, A. V., and Nechaev, A. A.: Linear recurring sequences over rings and modules. J. of Math. Sciences 76 (1995), No. 6, 2793–2915

    Article  MathSciNet  MATH  Google Scholar 

  10. Kuzmin, A. S., Kurakin, V. L., and Nechaev, A. A.: Pseudorandom and polylinear sequences. Proceedings in Discrete Math. 1 139–202. Moscow, TVP (1997) (in Russian)

    Google Scholar 

  11. Nathanson, M. B.: Difference operators and periodic sequences over finite modules. Acta Math. Acad. Hangar. 28 (1976), No. 3–4, 219–224

    Article  MathSciNet  MATH  Google Scholar 

  12. Nechaev, V. I.: Recurring sequences. Uchen. Zap. Mosk. Ped. Inst. 375 (1971) 103–123 (in Russian)

    Google Scholar 

  13. Nechaev, V. I.: Linear congruences modulo powers of a prime ideal and linear recurring sequences. Uchen. Zap. Mosk. Ped. Inst. 375 (1971) 124–132 (in Russian)

    Google Scholar 

  14. Nechaev, A. A.: Linear recurring sequences over commutative rings. Diskrete Math. and Appl. 2 (1992), No. 6 (VSP)

    Google Scholar 

  15. Nechaev, A. A.: Linear recurring sequences over quasi-Frobenius modules. Russian Math. Surveys 48 (1993), No. 3, 209–210

    Article  MathSciNet  Google Scholar 

  16. Nechaev A. A.: Finite Quasi-Frobenius modules, applications to codes and linear recurrences. Fundament. i Prikl. Matem. 1 (1995), No. 1, 229–254 (in Russian)

    MathSciNet  MATH  Google Scholar 

  17. Nechaev, A. A. and Tzypyshev, V. N.: Polylinear recurrences over bimodules. Math. methods and Appl. In: Proceedings of the III Math. School of Moscow State Social Univ., 1995 Jan. 24–29. 95–100

    Google Scholar 

  18. Nechaev, A. A.: Polylinear recurring sequences over modules and quasi-Frobenius modules. In: Proc. First Int. Tainan—Moscow Algebra Workshop, 1994. Walter de Gruyter & Co., Berlin—N. Y. (1996) 283–298

    Google Scholar 

  19. Nomura, T., Miyakawa H., Imai, H., and Fukuda, A.: A theory of two-dimensional linear recurring arrays. IEEE Trans. Inform. Theory 18 (1972), No. 6, 775–785

    Article  MathSciNet  MATH  Google Scholar 

  20. Sakata, S.: General theory of doubly periodic arrays over an arbitrary finite field and its applications. IEEE Trans. Inform. Theory 24 (1978) 719–730

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center of New Informational Technologies, Moscow State University, Russia

    V. L. Kurakin, A. V. Mikhalev & A. A. Nechaev

Authors
  1. V. L. Kurakin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Mikhalev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Nechaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIAFA, Université Paris VII, 2, Place Jussieu, 75251, Paris Cedex 05, France

    Daniel Krob

  2. Moscow State University, 119899, Moscow, Russia

    Alexander A. Mikhalev  & Alexander V. Mikhalev  & 

  3. The University of Hong Kong, Hong Kong

    Alexander A. Mikhalev

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kurakin, V.L., Mikhalev, A.V., Nechaev, A.A. (2000). Polylinear Recurring Sequences over a Bimodule. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds) Formal Power Series and Algebraic Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04166-6_46

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-04166-6_46

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08662-5

  • Online ISBN: 978-3-662-04166-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation