Probabilistic Existence Results for Separable Codes

Abstract

Separable codes were defined by Cheng and Miao in 2011, motivated by applications to the identification of pirates in a multimedia setting. Combinatorially, t-separable codes lie somewhere between tframeproof and (t − 1)-frameproof codes: all t-frameproof codes are t-separable, and all t-separable codes are (t − 1)-frameproof. Results for frameproof codes show that (when q is large) there are q-ary tseparable codes of length n with approximately q ⌈n/t⌉ codewords, and that no q-ary t-separable codes of length n can have more than approximately q ⌈n/(t−1)⌉ codewords.The paper provides improved probabilistic existence results for tseparable codes when t ≥ 3. More precisely, for all t ≥ 3 and all n ≥ 3, there exists a constant κ (depending only on t and n) such that there exists a q-ary t-separable code of length n with at least κq n/(t−1) codewords for all sufficiently large integers q. This shows, in particular, that the upper bound (derived from the bound on (t − 1)frameproof codes) on the number of codewords in a t-separable code is realistic.The results above are more surprising after examining the situation when t = 2. Results due to Gao and Ge show that a q-ary 2-separable code of length n can contain at most 3 2 q 2⌈n/3⌉ − 1 2 q ⌈n/3⌉ codewords, 1 and that codes with at least κq 2n/3 codewords exist. Thus optimal 2-separable codes behave neither like 2-frameproof nor 1-frameproof codes.The paper also observes that the bound of Gao and Ge can be strengthened to show that the number of codewords of a q-ary 2separable code of length n is at most q ⌈2n/3⌉ + 1 2 q ⌊n/3⌋ (q ⌊n/3⌋ − 1).

Asymptotically optimal 2 ¯ $\overline {2}$ -separable codes with length 4

“…S (1,1,1,2) = {(1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1)}; 1,1,3) = {(1, 1, 1, 3), (1, 1, 3, 1), (1, 3, 1, 1), (3, 1, 1, 1…”

“…. , c n ) ∈ C is Jing Jiang jjiang2008@hotmail.com Minquan Cheng chengqinshi@hotmail.com Xiaohu Tang xhutang@home.swjtu.edu.cn 1 Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, China 2 Guangxi Key Laboratory of Multi-source Information Mining & Security, Guangxi Normal University, Guilin, China called a codeword. Without loss of generality, we may assume Q = {0, 1, .…”

“…For more details on separable codes, the interested reader may refer to [2,[4][5][6][7]. Consider the case (t, n) = (2, 4).…”

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

“…Theorem 1.9. ( [4]) Let n and t be fixed integers such that n ≥ 2 and t ≥ 3. There exists a positive constant κ, depending only on n and t, so that there is a q-ary t-separable code of length n with at least κq n/(t−1) codewords for all sufficiently large integers q. Corollary 1.10.…”

“…in Example 3.2 can be regarded as a partial Latin square of order 4, which corresponds to an optimal 3-SC (3,12,4). In this section, we use partial Latin squares to construct 3-SC(3, M, q)s. The definition of a partial Latin square is given below.…”