Skip to main content
SpringerLink
Log in

(2, 1)-Separating systems beyond the probabilistic bound

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Building on previous results of Xing, we give new lower bounds on the rate of linear intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry (although nothing beyond the basic theory of linear systems on curves).

Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years.

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. N. Alon and J. H. Spencer, The Probabilistic Method, 2nd ed., Wiley-Interscience, New York 2000.

    Book  MATH  Google Scholar 

  2. S. R. Blackburn, Frameproof codes, SIAM Journal on Discrete Mathematics 16 (2003), 499–510.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Boneh and J. Shaw, Collusion-secure fingerprinting for digital data, IEEE Transactions on Information Theory 44 (1998), 1897–1905.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Cohen and A. Lempel, Linear intersecting codes, Discrete Mathematics 56 (1985), 35–43.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Cohen and H. G. Schaathun, Asymptotic Overview on Separating Codes, Reports in Informatics 248, University of Bergen, 2003.

  6. V. G. Drinfeld and S. G. Vladut, Number of points of an algebraic curve, Annals of Functional Analysis 17 (1983), 53–54.

    MathSciNet  Google Scholar 

  7. A. D. Friedman, R. L. Graham and J. D. Ullman, Universal single transition time asynchronous state assignments, IEEE Transactions on Computers Publication 18 (1969), 541–547.

    Article  Google Scholar 

  8. A. Garcia and H. Stichtenoth, A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound, Inventiones Mathematicae 121 (1995), 211–222.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 28 (1981), 721–724.

    MathSciNet  MATH  Google Scholar 

  10. J. Komlós, unpublished, cited in [15].

  11. J. Körner, On the extremal combinatorics of the Hamming space, Journal of Combinatorial Theory, Series A 71 (1995), 112–126.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Krasnopeev and Yu. L. Sagalovich, The Kerdock codes and separating systems, in Proceedings of the 8th Intrnational workshop on Algebraic and Combinatorial Coding Theory (ACCT-8), Tsarsloe Selo(Saint Petersburg), Russia, 2002, pp. 165–167.

  13. D. Laksov, Wronskians and Plücker formulas for linear systems on curves, Annales Scientifiques de l’École Normale Supérieure 17 (1984), 45–56.

    MathSciNet  MATH  Google Scholar 

  14. A. Lempel and S. Winograd, A new approach to error-correcting codes, IEEE Transactions on Information Theory 23 (1977), 503–508.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Miklós, Linear binary codes with intersection properties, Discrete Applied Mathematics 9 (1984), 187–196.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Neeman, Weierstrass points in characteristic p, Inventiones Mathematicae 75 (1984), 359–376.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Randriam, Hecke operators with odd determinant and binary frameproof codes beyond the probabilistic bound?, in Proceedings of 2010 IEEE Information Theory Workshop (ITW 2010 Dublin), to appear.

  18. H. Randriambololona, preliminary version of the present article: (2, 1)-separating systems beyond the probabilistic bound, April 2011, available online at: http://arxiv.org/abs/1010.5764v6

  19. H. Randriambololona, Diviseurs de la forme 2D-G sans sections et rang de la multiplication dans les corps finis, preprint, available online at: http://arxiv.org/abs/1103.4335

  20. A. Rényi, On random generating elements of a finite Boolean algebra, Acta Scientiarum Mathematicarum 22 (1961), 75–81.

    MATH  Google Scholar 

  21. Yu. L. Sagalovich, Cascade codes of automata states, Problemy Peredachi Informatsii 14 (1978), 77–85.

    MathSciNet  MATH  Google Scholar 

  22. Yu. L. Sagalovich and A. G. Chilingarjan, Separating systems and new scopes of its application, Information Processes 9 (2009), 225–248.

    Google Scholar 

  23. H. Stichtenoth, Algebraic Function Fields and Codes, Universitext, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  24. D. R. Stinson and R. Wei, Combinatorial properties and constructions of traceability schemes and frameproof codes, SIAM Journal on Discrete Mathematics 11 (1998), 41–53.

    Article  MathSciNet  MATH  Google Scholar 

  25. K.-O. Stöhr and F. Voloch, Weierstrass points and curves over finite fields, Proceedings of the London Mathematical Society 52 (1986), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. A. Tsfasman, S. G. Vladut and T. Zink, Modular curves, Shimura curves, and Goppa codes better than Varshamov-Gilbert bound, Mathematische Nachrichten 109 (1982), 21–28.

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Xing, Asymptotic bounds on frameproof codes, IEEE Transactions on Information Theory 48 (2002), 2991–2995.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. École nationale supérieure des télécommunications (“Telecom ParisTech”) & LTCI CNRS UMR 5141, 46 rue Barrault, 75634, Paris cedex 13, France

    Hugues Randriambololona

Authors
  1. Hugues Randriambololona
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hugues Randriambololona.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Randriambololona, H. (2, 1)-Separating systems beyond the probabilistic bound. Isr. J. Math. 195, 171–186 (2013). https://doi.org/10.1007/s11856-012-0126-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0126-9

Keywords

Search

Navigation