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Nonoverlapping Convex Polytopes with Vertices in a Boolean Cube and Other Problems in Coding Theory

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Abstract

We establish relations between several problems that are quite far from each other at first glance and formulate a number of open problems.

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References

  1. Cheng, M. and Miao, Y., On Anti-Collusion Codes and Detection Algorithms for Multimedia Fingerprinting, IEEE Trans. Inform. Theory, 2011, vol. 57, no. 7, pp. 4843–4851. https://doi.org/10.1109/TIT.2011.2146130

    Article  MathSciNet  MATH  Google Scholar 

  2. Egorova, E.E., and Kabatiansky, G.A., Separable Collusion-Secure Multimedia Codes, Probl. Peredachi Inf., 2021, vol. 57, no. 2, pp. 90–111 [Probl. Inf. Transm. (Engl. Transl.), 2021, vol. 57, no. 2, pp. 178–198]. https://doi.org/10.1134/S003294602102006X

    MathSciNet  MATH  Google Scholar 

  3. Boneh, D. and Shaw, J., Collusion-Secure Fingerprinting for Digital Data, IEEE Trans. Inform. Theory, 1998, vol. 44, no. 5, pp. 1897–1905. https://doi.org/10.1109/18.705568

    Article  MathSciNet  MATH  Google Scholar 

  4. 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. https://doi.org/10.1109/TIT.2003.809570

    Article  MathSciNet  MATH  Google Scholar 

  5. Erdős, P. and Turán, P., On a Problem of Sidon in Additive Number Theory, and on Some Related Problems, J. London Math. Soc., 1941, vol. 16, no. 4, pp. 212–215. https://doi.org/10.1112/jlms/s1-16.4.212

    Article  MathSciNet  MATH  Google Scholar 

  6. Babai, L. and Sós, V.T., Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs, European J. Combin., 1985, vol. 6, no. 2, pp. 101–114. https://doi.org/10.1016/S0195-6698(85)80001-9

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, G., Litsyn, S., and Zémor, G., Binary B2-Sequences: A New Upper Bound, J. Combin. Theory Ser. A, 2001, vol. 94, no. 1, pp. 152–155. https://doi.org/10.1006/jcta.2000.3127

    Article  MathSciNet  MATH  Google Scholar 

  8. Györfi, L., Győri, S., Laczay, B., and Ruszinkó, M., Lectures on Multiple Access Channels, book draft, 2005. Available at http://www.szit.bme.hu/~gyori/AFOSR_05/book.pdf

  9. Kabatiansky, G.A. and Lebedev, V.S., On Metric Dimension of Nonbinary Hamming Spaces, Probl. Peredachi Inf., 2018, vol. 54, no. 1, pp. 54–62 [Probl. Inf. Transm. (Engl. Transl.), 2018, vol. 54, no. 1, pp. 48–55]. https://doi.org/10.1134/S0032946018010040

    MathSciNet  MATH  Google Scholar 

  10. D’yachkov, A.G. and Rykov, V.V., On a Coding Model for a Multiple-Access Adder Channel, Probl. Peredachi Inf., 1981, vol. 17, no. 2, pp. 26–38 [Probl. Inf. Transm. (Engl. Transl.), 1981, vol. 17, no. 2, pp. 94–104]. http://mi.mathnet.ru/eng/ppi1390

    MathSciNet  MATH  Google Scholar 

  11. Wolf, J.K., Born Again Group Testing: Multiaccess Communications, IEEE Trans. Inform. Theory, 1985, vol. 31, no. 2, pp. 185–191. https://doi.org/10.1109/TIT.1985.1057026

    Article  MathSciNet  MATH  Google Scholar 

  12. Erdős, P. and Rényi, A., On Two Problems of Information Theory, Magyar Tud. Akad. Mat. Kutató Int. Közl., 1963, vol. 8, no. 1–2, pp. 229–243. Available at http://static.renyi.hu/renyi_cikkek/1963_on_two_problems_of_information_theory.pdf

    MathSciNet  MATH  Google Scholar 

  13. Lindström, B., On a Combinatory Detection Problem. I, Magyar Tud. Akad. Mat. Kutató Int. Közl., 1964, vol. 9, no. 1–2, pp. 195–207.

    MathSciNet  MATH  Google Scholar 

  14. Cantor, D.G. and Mills, W.H., Determination of a Subset from Certain Combinatorial Properties, Canad. J. Math., 1966, vol. 18, pp. 42–48. https://doi.org/10.4153/CJM-1966-007-2

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, Z. and Polyanskii, N., On the Metric Dimension of Cartesian Powers of a Graph, J. Combin. Theory Ser. A, 2019, vol. 165, pp. 1–14. https://doi.org/10.1016/j.jcta.2019.01.002

    Article  MathSciNet  MATH  Google Scholar 

  16. Bshouty, N.H. and Mazzawi, H., On Parity Check \((0,1)\)-Matrix over \(\mathbb{Z}_p\), in Proc. 22nd Annu. ACM–SIAM Symp. on Discrete Algorithms (SODA’11), San Francisco, CA, Jan. 23–25, 2011, pp. 1383–1394. https://dl.acm.org/doi/10.5555/2133036.2133142

  17. Fan, J., Gu, Y., Hachimori, M., and Miao, Y., Signature Codes for Weighted Binary Adder Channel and Multimedia Fingerprinting, IEEE Trans. Inform. Theory, 2021, vol. 67, no. 1, pp. 200–216. https://doi.org/10.1109/TIT.2020.3033445

    Article  MathSciNet  MATH  Google Scholar 

  18. Egorova, E.E., Fernandez, M., Kabatiansky, G.A., and Miao, Y., Existence and Construction of Complete Traceability Multimedia Fingerprinting Codes Resistant to Averaging Attack and Adversarial Noise, Probl. Peredachi Inf., 2020, vol. 56, no. 4, pp. 97–108 [Probl. Inf. Transm. (Engl. Transl.), 2000, vol. 56, no. 4, pp. 388–398]. https://doi.org/10.1134/S0032946020040080

    MathSciNet  MATH  Google Scholar 

  19. Vorobyev, I., Complete Traceability Multimedia Fingerprinting Codes Resistant to Averaging Attack and Adversarial Noise with Optimal Rate, Des. Codes Cryptogr., 2022, Open Access Article. https://doi.org/10.1007/s10623-022-01144-x

  20. Gkagkos, M., Pradhan, A.K., Amalladinne, V., Narayanan, K., Chamberland, J-F., and Georghiades, C.N., Approximate Support Recovery Using Codes for Unsourced Multiple Access, in Proc. 2021 IEEE Int. Symp. on Information Theory (ISIT’2021), Melbourne, Australia, July 12–20, 2021, pp. 2948–2953. https://doi.org/10.1109/ISIT45174.2021.9517995

  21. Donoho, D.L., Compressed Sensing, IEEE Trans. Inform. Theory, 2006, vol. 52, no. 4, pp. 1289–1306. https://doi.org/10.1109/TIT.2006.871582

    Article  MathSciNet  MATH  Google Scholar 

  22. Candès, E.J. and Tao, T., Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?, IEEE Trans. Inform. Theory, 2006, vol. 52, no. 12, pp. 5406–5425. https://doi.org/10.1109/TIT.2006.885507

    Article  MathSciNet  MATH  Google Scholar 

  23. Candès, E.J., Romberg, J.K., and Tao, T., Stable Signal Recovery from Incomplete and Inaccurate Measurements, Comm. Pure Appl. Math., 2006, vol. 59, no. 8, pp. 1207–1223. https://doi.org/10.1002/cpa.20124

    Article  MathSciNet  MATH  Google Scholar 

  24. Wen, J., Zhou, Z., Wang, J., Tang, X., and Mo, Q., A Sharp Condition for Exact Support Recovery with Orthogonal Matching Pursuit, IEEE Trans. Signal Process., 2017, vol. 65, no. 6, pp. 1370–1382. https://doi.org/10.1109/TSP.2016.2634550

    Article  MathSciNet  MATH  Google Scholar 

  25. Mehrabi, M. and Tchamkerten, A., Error-Correction for Sparse Support Recovery Algorithms, in Proc. 2021 IEEE Int. Symp. on Information Theory (ISIT’2021), Melbourne, Australia, July 12–20, 2021, pp. 1754–1759. https://doi.org/10.1109/ISIT45174.2021.9518027

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Funding

The research of G.A. Kabatiansky was was carried out at the expense of the Russian Science Foundation, project no. 22-41-02028.

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Translated from Problemy Peredachi Informatsii, 2022, Vol. 58, No. 4, pp. 50–61. https://doi.org/10.31857/S0555292322040052

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Janabekova, A., Kabatiansky, G., Kamel, I. et al. Nonoverlapping Convex Polytopes with Vertices in a Boolean Cube and Other Problems in Coding Theory. Probl Inf Transm 58, 341–351 (2022). https://doi.org/10.1134/S0032946022040056

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  • DOI: https://doi.org/10.1134/S0032946022040056

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