Skip to main content
SpringerLink
Log in

Fast skew-feedback shift-register synthesis

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

A skew-feedback shift-register is a generalization of a linear-feedback shift-register and can be applied in decoding (interleaved) Reed–Solomon codes or Gabidulin codes beyond half their code distance. A fast algorithm is proposed which synthesizes all shortest skew-feedback shift-registers generating L sequences of varying length over a field. For fixed L, the time complexity of the algorithm is \({{\mathcal O}(M(N) \log N)}\) operations, where N is the length of a longest sequence and M(N) is the complexity of the multiplication of two skew polynomials of maximum degree N.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blahut R.E.: Theory and practice of error control codes. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  2. Ore O.: Theory of non-commutative polynomials. Ann. Math. 34, 480–508 (1933)

    Article  MATH  MathSciNet  Google Scholar 

  3. Hassan Y., Sidorenko V.: Fast recursive linearized feedback shift register synthesis In: 12th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT-12), Akademgorodok, Novosibirsk (2010).

  4. Shönhage A., Strassen V.: Schnelle Multiplikation großer Zahlen”. Computing 7, 281–292 (1971)

    Article  Google Scholar 

  5. Sidorenko V., Jiang L., Bossert M.: Skew-feedback shift-register synthesis and decoding interleaved Gabidulin codes. IEEE Trans. Inf. Theory IT-57, 621–632 (2011)

    Article  MathSciNet  Google Scholar 

  6. Wachter A., Afanassiev V., Sidorenko V.: Fast decoding of Gabidulin codes. The 7th International Workshop on Coding and Cryptography 2011, pp. 433–442. Paris (2011).

  7. Zeh A., Wachter A.: Fast multi-sequence shift-register synthesis with the Euclidean algorithm. Adv. Math. Commun. 5(4), 667–680 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Schmidt G., Sidorenko V.R., Sidorenko V.R.: Collaborative decoding of interleaved Reed–Solomon codes and concatenated code designs. IEEE Trans. Inf. Theory 55, 2991–3012 (2009)

    Article  Google Scholar 

  9. Schmidt G., Sidorenko V.R., Bossert M.: Syndrome decoding of Reed–Solomon codes beyond half the minimum distance based on shift- register synthesis. IEEE Trans. Inf. Theory 56(10), 5245–5252 (2010)

    Article  MathSciNet  Google Scholar 

  10. Berlekamp E.R.: Algebraic Coding Theory. McGraw–Hill, New York (1968)

    MATH  Google Scholar 

  11. Massey J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory IT-15, 122–127 (1969)

    Article  MathSciNet  Google Scholar 

  12. Sugiyama Y., Kasahara M., Hirasawa S., Namekawa T.: A method for solving key equations for decoding Goppa codes. Inf. Control 27, 87–99 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mills W.H.: Continued fractions and linear recurrences. Math. Comput. 29(129), 173–180 (1975)

    Article  MATH  Google Scholar 

  14. Feng G.-L., Tzeng K.K.: A generalization of the Berlekamp–Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes. IEEE Trans. Inf. Theory IT-37, 1274–1287 (1991)

    Article  MathSciNet  Google Scholar 

  15. Feng G.-L., Tzeng K.K.: A generalized Euclidean algorithm for multisequence shift-register synthesis. IEEE Trans. Inf. Theory 35(3), 584–594 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Wang L.-P., Zhu Y.-F., Pei D.-Y.: On the lattice basis reduction multisequence synthesis algorithm. IEEE Trans. Inf. Theory IT-50, 2905–2910 (2004)

    Article  MathSciNet  Google Scholar 

  17. Kurakin V.L.: The Berlekamp–Massey algorithm over a finite commutative ring, Probl. Inf. Transm. 35(2), 125–135 (1999)

    MATH  MathSciNet  Google Scholar 

  18. Kuijper M.: An algorithm for constructing a minimal partial realization in the multivariable case. Syst. Control Lett. 31(4), 225–233 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sakata S.: The BM algorithm and the BMS algorithm. In: Vardy A. (ed.) Codes, Curves and Signals: Common Threads in Communications, pp. 39–52. Boston, Kluwer Academic Publishers, ISBN 0-792-38374-5, (1998).

  20. Sidorenko V., Schmidt G.: A linear algebraic approach to multisequence shift-register synthesis. Probl. Inf. Transm. 47, 149–165 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wang L.P., Wang Q.L., Wang K.P.: A lattice-based shift-register synthesis for multi-sequences of varying length. In: Proceedings of the 2008 IEEE International Symposium on Information Theory (ISIT 2008), 6–11 July, pp. 1751–1754. Toronto, Canada (2008).

  22. Gabidulin E.M.: Theory of codes with maximal rank distance. Probl. Inf. Transm. (English translation). 21, 1–12 (1985)

    MATH  Google Scholar 

  23. Paramonov A.V., Tretjakov O.V.: An analogue of Berlekamp–Massey algorithm for decoding codes in rank metric. In: Proceedings of Moscow Institute for Physics and Technology (MIPT), Moscow, Russia (In Russian) (1991).

  24. Richter G., Plass S.: Error and erasure decoding of rank-codes with a modified Berlekamp–Massey algorithm. Proceedings of ITG Conference on Source and Channel Coding, pp. 249–256. Erlangen, Germany (2004).

  25. Sidorenko V., Richter G., Bossert M.: Linearized shift-register synthesis. IEEE Trans. Inf. Theory IT-57(9), 6025–6032 (2011)

    Article  MathSciNet  Google Scholar 

  26. Wachter A., Sidorenko V., Bossert M.: A fast linearized Euclidean algorithm for decoding Gabidulin codes. In: Proceedings of 12th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT 2012), pp. 298–303. Akademgorodok, Novosibirsk, Russia (2010).

  27. Sidorenko V., Bossert M.: Fast skew-feedback shift-register synthesis. In: Borges J., Villanueva M. (eds.) 3rd International Castle meeting on Coding Theory and Applications, pp. 267–272. Spain, Cardona (2011).

Download references

Author information

Authors and Affiliations

  1. Institute of Communications Engineering, Ulm University, Ulm, Germany

    Vladimir Sidorenko & Martin Bossert

Authors
  1. Vladimir Sidorenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Bossert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Sidorenko.

Additional information

Vladimir Sidorenko is on leave from IITP Russian Academy of Sciences.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding Theory and Applications”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sidorenko, V., Bossert, M. Fast skew-feedback shift-register synthesis. Des. Codes Cryptogr. 70, 55–67 (2014). https://doi.org/10.1007/s10623-012-9663-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-012-9663-9

Keywords

Mathematics Subject Classification

Search

Navigation