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Bounds and constructions for \(\overline {3}\)-strongly separable codes with length 3

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Abstract

Separable code (SC, Cheng and Miao IEEE Trans. Inf. Theory 57, 4843–4851, 2011), frameproof code (FPC, Boneh and Shaw IEEE Trans. Inf. Theory 44, 1897–1905, 1998) and strongly separable code (SSC, Jiang et al. Des. Codes Cryptogr. 79:303–318, 2016) are used to construct anti-collusion codes. SSC is better than FPC and SC in the applications for multimedia fingerprinting since SSC has lower identifying complexity than that of SC (the same complexity as FPC) and weaker structure than that of FPC. In this paper, we first derive several upper bounds on the number of codewords of a \(\overline {t}\)-SSC. Then we focus on \(\overline {3}\)-SSCs with codeword length 3 and obtain the following two main results: (1) An equivalence between an SSC and an SC is derived; (2) An improved lower bound Ω(q 5/3 + q 4/3 − q) on the size of a q-ary SSC when \(q={q_{1}^{6}}\) for any prime power q 1 ≡ 1 (mod 6), which is better than the previously known bound \(\lfloor \sqrt {q}\rfloor ^{3}\), is obtained by means of a difference matrix and a known result on the subsets of \(\mathbb {F}^{n}_{q}\) containing no three points on a line.

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Acknowledgements

The authors express their sincere thanks to the two anonymous reviewers for their critical comments and suggestions which made this paper significantly improved, and to the editor for his/her excellent editorial job. This work is in part supported by NSFC (No.11301098, 11601096 and 61672176), 2016GXNSF(No. FA380009 and CA380021), 2014GXNSFDA118001, Guangxi Higher Institutions’ Program of Introducing 100 High-Level Overseas Talents, Guangxi Collaborative Innovation Center of Multi-source Information Integration and Intelligent Processing, and Guangxi Bagui Scholar Teams for Innovation and Research.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, China

    Xuli Zhang

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

    Jing Jiang & Minquan Cheng

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  1. Xuli Zhang
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  2. Jing Jiang
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  3. Minquan Cheng
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Correspondence to Minquan Cheng.

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Zhang, X., Jiang, J. & Cheng, M. Bounds and constructions for \(\overline {3}\)-strongly separable codes with length 3. Cryptogr. Commun. 10, 555–565 (2018). https://doi.org/10.1007/s12095-017-0235-8

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  • DOI: https://doi.org/10.1007/s12095-017-0235-8

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