Bounds and constructions for 3 ¯ $\overline {3}$ -strongly separable codes with length 3
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.
Publisher URL: https://link.springer.com/article/10.1007/s12095-017-0235-8
DOI: 10.1007/s12095-017-0235-8
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