Problems of Information Transmission

, Volume 53, Issue 1, pp 30–41 | Cite as

Bounds on the rate of separating codes

Coding Theory


A code with words in a finite alphabet is said to be an (s, l) separating code if for any two disjoint collections of its words of size at most s and l, respectively, there exists a coordinate in which the set of symbols of the first collection do not intersect the set of symbols of the second. The main goal of the paper is obtaining new bounds on the rate of (s, l) separating codes. Bounds on the rate of binary (s, l) separating codes, the most important for applications, are studied in more detail. We give tables of numerical values of the best presently known bounds on the rate.


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  1. 1.
    Friedman, A.D., Graham, R.L., and Ullman, J.D., Universal Single Transition Time Asynchronous State Assignments, IEEE Trans. Comp., 1969, vol. 18, no. 6, pp. 541–547.CrossRefGoogle Scholar
  2. 2.
    Sagalovich, Yu.L., Completely Separating Systems, Probl. Peredachi Inf., 1982, vol. 18, no. 2, pp. 74–82 [Probl. Inf. Trans. (Engl. Transl.), 1982, vol. 18, no. 2, pp. 140–146].MathSciNetMATHGoogle Scholar
  3. 3.
    Sagalovich, Yu.L., Separating Systems, Probl. Peredachi Inf., 1994, vol. 30, no. 2, pp. 14–35 [Probl. Inf. Trans. (Engl. Transl.), 1994, vol. 30, no. 2, pp. 105–123].MathSciNetMATHGoogle Scholar
  4. 4.
    Stinson, D.R., Wei, R., and Chen, K., On Generalized Separating Hash Families, J. Combin. Theory Ser. A, 2008, vol. 115, no. 1, pp. 105–120.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boneh, D. and Shaw, J., Collusion-Secure Fingerprinting for Digital Data, IEEE Trans. Inform. Theory, 1998, vol. 44, no. 5, pp. 1897–1905.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Staddon, J.N., Stinson, D.R., and Wei, R., Combinatorial Properties of Frameproof and Traceability Codes, IEEE Trans. Inform. Theory, 2001, vol. 47, no. 3, pp. 1042–1049.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Barg, A., Blakley, G.R., and Kabatiansky, G.A., Digital Fingerprinting Codes: Problem Statements, Constructions, Identification of Traitors, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 4, pp. 852–865.CrossRefMATHGoogle Scholar
  8. 8.
    Barg, A. and Kabatiansky, G., Robust Parent-Identifying Codes and Combinatorial Arrays, IEEE Trans. Inform. Theory, 2013, vol. 59, no. 2, pp. 994–1003.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Magó, G., Monotone Functions in Sequential Circuits, IEEE Trans. Comput., 1973, vol. 22, no. 10, pp. 928–933.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Mitchell, C.J. and Piper, F.C., Key Storage in Secure Networks, Discrete Appl. Math., 1988, vol. 21, no. 3, pp. 215–228.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cohen, G.D. and Schaathun, H.G., Asymptotic Overview on Separating Codes, Tech. Rep. of Dept. of Informatics, Univ. of Bergen, Bergen, Norway, May 2003, no.248.Google Scholar
  12. 12.
    D’yachkov, A., Vilenkin, P., Macula, A., and Torney, V., Families of Finite Sets inWhich No Intersection of Sets Is Covered by the Union of s Others, J. Combin. Theory Ser. A, 2002, vol. 99, no. 2, pp. 195–218.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Sagalovich, Yu.L., Concatenated Codes of Automaton States, Probl. Peredachi Inf., 1978, vol. 14, no. 2, pp. 77–85 [Probl. Inf. Trans. (Engl. Transl.), 1978, vol. 14, no. 2, pp. 132–138].MathSciNetGoogle Scholar
  14. 14.
    D’yachkov, A.G., Vorob’ev, I.V., Polyansky, N.A., and Shchukin, V.Yu., Bounds on the Rate of Disjunctive Codes, Probl. Peredachi Inf., 2014, vol. 50, no. 1, pp. 31–63 [Probl. Inf. Trans. (Engl. Transl.), 2014, vol. 50, no. 1, pp. 27–56].MathSciNetMATHGoogle Scholar
  15. 15.
    Shangguan, C., Wang, X., Ge, G., and Miao, Y., New Bounds for Frameproof Codes, arXiv:1411.5782 [cs.IT], 2014.Google Scholar
  16. 16.
    D’yachkov, A.G., Rykov, V.V., Deppe, C., and Lebedev, V.S., Superimposed Codes and Threshold Group Testing, Information Theory, Combinatorics, and Search Theory. In Memory of Rudolf Ahlswede, Aydinian, H.K., Cicalese, F., and Deppe, C., Eds., Lect. Notes Comp. Sci., vol. 7777, Berlin: Springer, 2013, pp. 509–533.CrossRefGoogle Scholar
  17. 17.
    Engel, K., Interval Packing and Covering in the Boolean Lattice, Combin. Probab. Comput., 1996, vol. 5, no. 4, pp. 373–384.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lebedev, V.S., Asymptotic Upper Bound for the Rate of (w, r) Cover-Free Codes, Probl. Peredachi Inf., 2003, vol. 39, no. 4, pp. 3–9 [Probl. Inf. Trans. (Engl. Transl.), 2003, vol. 39, no. 4, pp. 317–323].MathSciNetMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2017

Authors and Affiliations

  1. 1.Probability Theory ChairFaculty of Mechanics and Mathematics, Lomonosov Moscow State UniversityMoscowRussia

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