Designs, Codes and Cryptography

, Volume 81, Issue 2, pp 317–335

# Bounds and constructions for $${\overline{3}}$$-separable codes with length 3

• Minquan Cheng
• Jing Jiang
• Haiyan Li
• Ying Miao
• Xiaohu Tang
Article

## Abstract

Separable codes were introduced to provide protection against illegal redistribution of copyrighted multimedia material. Let $${\mathcal {C}}$$ be a code of length n over an alphabet of q letters. The descendant code $${\mathsf{desc}}({\mathcal {C}}_0)$$ of $${\mathcal {C}}_0 = \{\mathbf{c}_1, \mathbf{c}_2, \ldots , \mathbf{c}_t\} \subseteq {{\mathcal {C}}}$$ is defined to be the set of words $$\mathbf{x} = (x_1, x_2, \ldots ,x_n)^T$$ such that $$x_i \in \{c_{1,i}, c_{2,i}, \ldots , c_{t,i}\}$$ for all $$i=1, \ldots , n$$, where $$\mathbf{c}_j=(c_{j,1},c_{j,2},\ldots ,c_{j,n})^T$$. $${\mathcal {C}}$$ is a $${\overline{t}}$$-separable code if for any two distinct $${\mathcal {C}}_1, {\mathcal {C}}_2 \subseteq {\mathcal {C}}$$ with $$|{\mathcal {C}}_1| \le t$$, $$|{\mathcal {C}}_2| \le t$$, we always have $${\mathsf{desc}}({\mathcal {C}}_1) \ne {\mathsf{desc}}({\mathcal {C}}_2)$$. Let $$M({\overline{t}},n,q)$$ denote the maximal possible size of such a separable code. In this paper, an upper bound on $$M({\overline{3}},3,q)$$ is derived by considering an optimization problem, and then two constructions for $${\overline{3}}\hbox {-SC}(3,M,q)$$s are provided by means of perfect hash families and Steiner triple systems.

## Keywords

Multimedia fingerprinting Separable code Partial Latin square Perfect hash family Steiner triple system

## Mathematics Subject Classification

94A62 94B25 05B15 05B30

## Notes

### Acknowledgments

Cheng is supported in part by NSFC (No.11301098), Guangxi Higher Institutions’ Program of Introducing 100 High-Level Overseas Talents, and Guangxi “Bagui Scholar” Teams for Innovation and Research. Jiang is supported by Guangxi Natural Science Foundations (No.2013GXNSFCA019001 and 2014GXNSFDA118001), and Foundation of Guangxi Education Department (No.2013YB039). Miao is supported by JSPS Grant-in-Aid for Scientific Research (C) (No.15K04974)

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## Authors and Affiliations

• Minquan Cheng
• 1
• Jing Jiang
• 2
• Haiyan Li
• 3
• Ying Miao
• 4
• Xiaohu Tang
• 5
1. 1.Information Security and National Computing Grid LaboratorySouthwest Jiaotong UniversityChengduChina
2. 2.School of Computer Science and Information TechnologyGuangxi Normal UniversityGuilinChina
3. 3.School of Mathematics and StatisticsGuangxi Normal UniversityGuilinChina
4. 4.Faculty of Engineering, Information and SystemsUniversity of TsukubaTsukubaJapan
5. 5.Information Security and National Computing Grid LaboratorySouthwest Jiaotong UniversityChengduChina