Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
129 tokens/sec
GPT-4o
28 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Improved upper bounds for wide-sense frameproof codes (2402.05596v1)

Published 8 Feb 2024 in math.CO, cs.IT, and math.IT

Abstract: Frameproof codes have been extensively studied for many years due to their application in copyright protection and their connection to extremal set theory. In this paper, we investigate upper bounds on the cardinality of wide-sense $t$-frameproof codes. For $t=2$, we apply results from Sperner theory to give a better upper bound, which significantly improves a recent bound by Zhou and Zhou. For $t\geq 3$, we provide a general upper bound by establishing a relation between wide-sense frameproof codes and cover-free families. Finally, when the code length $n$ is at most $\frac{15+\sqrt{33}}{24}(t-1)2$, we show that a wide-sense $t$-frameproof code has at most $n$ codewords, and the unique optimal code consists of all weight-one codewords. As byproducts, our results improve several best known results on binary $t$-frameproof codes.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (32)
  1. Digital fingerprinting codes: problem statements, constructions, identification of traitors. IEEE Trans. Inform. Theory, 49(4):852–865, 2003.
  2. M. Bazrafshan and T. van Trung. Bounds for separating hash families. J. Combin. Theory Ser. A, 118(3):1129–1135, 2011.
  3. S. R. Blackburn. Combinatorial schemes for protecting digital content. In Surveys in combinatorics, 2003 (Bangor), volume 307 of London Math. Soc. Lecture Note Ser., pages 43–78. Cambridge Univ. Press, Cambridge, 2003.
  4. S. R. Blackburn. Frameproof codes. SIAM J. Discrete Math., 16(3):499–510, 2003.
  5. D. Boneh and J. Shaw. Collusion-secure fingerprinting for digital data. IEEE Trans. Inform. Theory, 44(5):1897–1905, 1998.
  6. Y. M. Chee and X. Zhang. Improved constructions of frameproof codes. IEEE Trans. Inform. Theory, 58(8):5449–5453, 2012.
  7. M. Cheng and Y. Miao. On anti-collusion codes and detection algorithms for multimedia fingerprinting. IEEE Trans. Inform. Theory, 57(7):4843–4851, 2011.
  8. P. Delsarte. Four fundamental parameters of a code and their combinatorial significance. Information and Control, 23:407–438, 1973.
  9. Bounds on the length of disjunctive codes. Problemy Peredachi Informatsii, 18(3):7–13, 1982 (Russian).
  10. Families of finite sets in which no set is covered by the union of r𝑟ritalic_r others. Israel J. Math., 51(1-2):79–89, 1985.
  11. Z. Füredi. On r𝑟ritalic_r-cover-free families. J. Combin. Theory Ser. A, 73(1):172–173, 1996.
  12. Some intriguing upper bounds for separating hash families. Science China Mathematics, 62:269–282, 2019.
  13. On tight bounds for binary frameproof codes. Des. Codes Cryptogr., 77(2-3):301–319, 2015.
  14. Non-adaptive hypergeometric group testing. Studia Scientiarum Mathematicarum Hungarica, 22:257–263, 1987.
  15. G. O. H. Katona. Intersection theorems for systems of finite sets. Acta Mathematica Academiae Scientiarum Hungarica, 15(3):329–337, 1964.
  16. W. Kautz and R. Singleton. Nonrandom binary superimposed codes. IEEE Trans. Inform. Theory, 10(4):363–377, 1964.
  17. D. J. Kleitman. On a combinatorial conjecture of Erdős. J. Combinatorial Theory, 1:209–214, 1966.
  18. D. Lubell. A short proof of Sperner’s Lemma. J. Combinatorial Theory, 1:299, 1966.
  19. L. D. Mešalkin. A generalization of Sperner’s theorem on the number of subsets of a finite set. Teor. Verojatnost. i Primenen, 8:219–220, 1963 (Russian, with German summary).
  20. D. T. Nagy and B. Patkós. On L𝐿Litalic_L-close Sperner systems. Graphs Combin., 37:789–796, 2021.
  21. A. Panoui, Wide-sense fingerprinting codes and honeycomb arrays. Ph.D. dissertation, Royal Holloway, University of London, 2012.
  22. M. Ruszinkó. On the upper bound of the size of the r𝑟ritalic_r-cover-free families. J. Combin. Theory Ser. A, 66(2):302–310, 1994.
  23. New bounds for frameproof codes. IEEE Trans. Inform. Theory, 63(11):7247–7252, 2017.
  24. C. Shangguan and G. Ge. New bounds on the number of tests for disjunct matrices. IEEE Trans. Inform. Theory, 62(12):7518–7521, 2016.
  25. E. Sperner. Ein satz über untermengen einer endlichen menge. Math. Z., 27(1):544–548, 1928.
  26. J. N. Staddon, D. R. Stinson and R. Wei. Combinatorial properties of frameproof and traceability codes. IEEE Trans. Inform. Theory, 47(3):1042–1049, 2001.
  27. D. R. Stinson and R. Wei. Combinatorial properties and constructions of traceability schemes and frameproof codes. SIAM J. Discrete Math., 11(1):41–53, 1998.
  28. T. van Trung. A tight bound for frameproof codes viewed in terms of separating hash families. Des. Codes Cryptogr., 72(3):713–718, 2014.
  29. R. Wei. On cover-free families. arXiv preprint, arXiv:2303.17524, 2023.
  30. Z. Xu and C. H. Yip. Sperner systems with restricted differences. arXiv preprint, arXiv:2210.02409, 2022.
  31. K. Yamamoto. Logarithmic order of free distributive lattice. J. Math. Soc. Japan, 6:343–353, 1954.
  32. J. Zhou and W. Zhou. Wide-sense 2-frameproof codes. Des. Codes Cryptogr., 88(12):2507–2519, 2020.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now