Skip to main content
SpringerLink
Log in

Almost cover-free codes

  • Coding Theory
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

We say that an s-subset of codewords of a code X is (s, l)-bad if X contains l other codewords such that the conjunction of these l words is covered by the disjunction of the words of the s-subset. Otherwise, an s-subset of codewords of X is said to be (s, l)-bad. A binary code X is called a disjunctive (s, l) cover-free (CF) code if X does not contain (s, l)-bad subsets. We consider a probabilistic generalization of (s, l) CF codes: we say that a binary code is an (s, l) almost cover-free (ACF) code if almost all s-subsets of its codewords are (s, l)-good. The most interesting result is the proof of a lower and an upper bound for the capacity of (s, l) ACF codes; the ratio of these bounds tends as s→∞ to the limit value log2 e/(le).

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. Kautz, W.H. and Singleton, R.C., Nonrandom Binary Superimposed Codes, IEEE Trans. Inform. Theory, 1964, vol. 10, no. 4, pp. 363–377.

    Article  MATH  Google Scholar 

  2. D’yachkov, A.G. and Rykov, V.V., Bounds on the Length of Disjunctive Codes, Probl. Peredachi Inf., 1982, vol. 18, no. 3, pp. 7–13 [Probl. Inf. Trans. (Engl. Transl.), 1982, vol. 18, no. 3, pp. 166–171].

    MathSciNet  MATH  Google Scholar 

  3. 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].

    MathSciNet  MATH  Google Scholar 

  4. Mitchell, C.J. and Piper, F.C., Key Storage in Secure Networks, Discrete Appl. Math., 1988, vol. 21, no. 3, pp. 215–228.

    Article  MathSciNet  MATH  Google Scholar 

  5. Sidelnikov, V.M. and Prikhodov, O.Yu., On the Construction of (w, r) Cover-Free Codes, Probl. Peredachi Inf., 2009, vol. 45, no. 1, pp. 36–40 [Probl. Inf. Trans. (Engl. Transl.), 2009, vol. 45, no. 1, pp. 32–36].

    MathSciNet  MATH  Google Scholar 

  6. 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].

    Google Scholar 

  7. 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.

    Article  MathSciNet  MATH  Google Scholar 

  8. D’yachkov, A.G., Vorobyev, I.V., Polyanskii, N.A., and Shchukin, V.Yu., Almost Disjunctive List- Decoding Codes, Probl. Peredachi Inf., 2015, vol. 51, no. 2, pp. 27–49 [Probl. Inf. Trans. (Engl. Transl.), 2015, vol. 51, no. 2, pp. 110–131].

    MathSciNet  Google Scholar 

  9. D’yachkov, A.G., Macula, A.J., and Rykov, V.V., New Applications and Results of Superimposed Code Theory Arising from Potentialities of Molecular Biology, Numbers, Information, and Complexity (Bielefeld, 1998), Althöfer, I., Cai, N., Dueck, G., Khachatrian, L., Pinsker, M.S., Sárközy, A., Wegener, I., and Zhang, Z., Eds., Boston: Kluwer, 2000, pp. 265–282.

  10. Bassalygo, L.A. and Rykov, V.V., Multiple-Access Hyperchannel, Probl. Peredachi Inf., 2013, vol. 49, no. 4, pp. 3–12 [Probl. Inf. Trans. (Engl. Transl.), 2013, vol. 49, no. 4, pp. 299–307].

    MathSciNet  MATH  Google Scholar 

  11. Gallager, R.G., Information Theory and Reliable Communication, New York: Wiley, 1968.

    MATH  Google Scholar 

  12. Csiszar, I. and Körner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems, New York: Academic; Budapest: Akad. Kiadó, 1981. Translated under the title Teoriya informatsii: teoremy kodirovaniya dlya diskretnykh sistem bez pamyati, Moscow: Mir, 1985.

    MATH  Google Scholar 

  13. D’yachkov, A.G., Vorobyev, I.V., Polyanskii, N.A., and Shchukin, V.Yu., Cover-Free Codes and Separating System Codes, in Proc. 2015 IEEE Int. Sympos. on Information Theory (ISIT’2015), Hong Kong, China, June 14–19, 2015, pp. 2894–2898.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    N. A. Polyansky

  2. Probability Theory Department, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia

    N. A. Polyansky

Authors
  1. N. A. Polyansky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Polyansky.

Additional information

Original Russian Text cN.A. Polyansky, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 2, pp. 46–60.

The research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Polyansky, N.A. Almost cover-free codes. Probl Inf Transm 52, 142–155 (2016). https://doi.org/10.1134/S0032946016020046

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946016020046

Search

Navigation