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Improved bounds on 2-frameproof codes with length 4

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Abstract

Frameproof codes (FPCs) are widely studied as they are classic fingerprinting codes that can protect copyrighted materials. The main interests are construction methods and bounds of the number of codewords of FPCs for a fixed length when the alphabet size approaches infinity. In this paper, we focus on the upper bound of the size of FPCs when the fixed length is 4 and the strength is 2. We obtain an upper bound \(2q^2-2q+7\) on the size of a q-ary 2-FPC of length 4 for any positive integer \(q> 48\). The best previously well known bound of this type of FPCs is \(2q^2-2\), which is due to Blackburn (SIAM J Discret Math 16:499–510, 2003). Our new upper bound improves the previous upper bound and it is not very far from the current best lower bound \(2q^2-4q+3\) obtained from the explicit construction due to Chee and Zhang (IEEE Trans Inf Theory 58:5449–5453, 2012).

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Acknowledgements

The authors express their sincere thanks to the Associate Editor, Prof. Ayshwarya Ganesan and three anonymous reviewers for their helpful suggestions which improved this paper significantly. Minquan Cheng and Jing Jiang were in part supported by NSFC (Nos. 11301098, 11601096), 2016GXNSF (Nos. FA380011, CA380021), Guangxi Higher Institutions Program of Introducing 100 High-Level Overseas Talents, Research Fund of Guangxi Key Lab of Multi-source Information Mining & Security (16-B-01, 18-B-01). The research of Qiang Wang was partially supported by NSERC of Canada.

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Authors and Affiliations

  1. Guangxi Key Lab of Multi-source Information Mining & Security, Guangxi Normal University, Guilin, China

    Minquan Cheng & Jing Jiang

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada

    Qiang Wang

Authors
  1. Minquan Cheng
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  2. Jing Jiang
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  3. Qiang Wang
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Correspondence to Minquan Cheng.

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Communicated by C. J. Colbourn.

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Cheng, M., Jiang, J. & Wang, Q. Improved bounds on 2-frameproof codes with length 4. Des. Codes Cryptogr. 87, 97–106 (2019). https://doi.org/10.1007/s10623-018-0490-5

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  • DOI: https://doi.org/10.1007/s10623-018-0490-5

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