可分码的概率存在性结果
Probabilistic existence results for separable codes
摘要：
Cheng和Miao在2011年定义了可分离码，其动机是在多媒体环境中识别盗版。组合地，$\overline{t}$可分离码位于$t$防帧码和$(t-1)$防帧码之间的某处：所有$t$防帧码都是$\overline{t}$可分离码，并且所有$\overline{t}$可分离码都是$(t-1)$防帧码.防帧码的结果表明（当$q$很大时）存在长度为$n$的$q$ -元$\overline{t}$ -可分码，其码字近似为$q^{\lceil n/t\rceil}$，并且长度为$n$的$q$ -元$\overline{t}$ -可分码不能有多于近似$q^{\lceil n/(t-1)\rceil}$码字.本文给出了当$t\geq 3$时，$\overline{t}$ -可分码的概率存在性的改进结果.更确切地说，对于所有$t\geq 3$和所有$n\geq 3$，存在一个常数$κ$（仅取决于$t$和$n$），使得对于所有足够大的整数$q$，存在一个长度为$n$的$q$ -元$\overline{t}$ -可分离码，该码至少具有$κq^{n/(t-1)}$码字.这表明，特别是，上界（从$(t-1)$防帧码的界限）的码字的数量在一个$\overline{t}$可分离的代码是现实的。上述结果是更令人惊讶的检查后，$t=2$的情况。Gao和Ge的结果表明，长度为$n$的$q$ -元$\overline{2}$ -可分码最多可以包含$\frac{3}{2}q^{2\lceil n/3\rceil}-\frac{1}{2}q^{\lceil n/3\rceil}$码字，并且存在至少包含$κq^{2n/3}$码字的码.因此，最优的$\overline{2}$ -可分离码既不像$2$ -防帧码，也不像$1$ -防帧码.同时，Gao-Ge界也得到了加强，证明了长度为$n$的$q$元$\overline{2}$可分码最多只能有\[ q^{\lceil 2n/3\rceil}+\tfrac{1}{2}q^{\lfloor n/3\rfloor}(q^{\lfloor n/3\rfloor}-1) \]码字.
英文摘要：
Separable codes were defined by Cheng and Miao in 2011, motivated by applications to the identification of pirates in a multimedia setting. Combinatorially, $\overline{t}$-separable codes lie somewhere between $t$-frameproof and $(t-1)$-frameproof codes: all $t$-frameproof codes are $\overline{t}$-separable, and all $\overline{t}$-separable codes are $(t-1)$-frameproof. Results for frameproof codes show that (when $q$ is large) there are $q$-ary $\overline{t}$-separable codes of length $n$ with approximately $q^{\lceil n/t\rceil}$ codewords, and that no $q$-ary $\overline{t}$-separable codes of length $n$ can have more than approximately $q^{\lceil n/(t-1)\rceil}$ codewords. The paper provides improved probabilistic existence results for $\overline{t}$-separable codes when $t\geq 3$. More precisely, for all $t\geq 3$ and all $n\geq 3$, there exists a constant $κ$ (depending only on $t$ and $n$) such that there exists a $q$-ary $\overline{t}$-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 $\overline{t}$-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 $\overline{2}$-separable code of length $n$ can contain at most $\frac{3}{2}q^{2\lceil n/3\rceil}-\frac{1}{2}q^{\lceil n/3\rceil}$ codewords, and that codes with at least $κq^{2n/3}$ codewords exist. So optimal $\overline{2}$-separable codes behave neither like $2$-frameproof nor $1$-frameproof codes. Also, the Gao--Ge bound is strengthened to show that a $q$-ary $\overline{2}$-separable code of length $n$ can have at most \[ q^{\lceil 2n/3\rceil}+\tfrac{1}{2}q^{\lfloor n/3\rfloor}(q^{\lfloor n/3\rfloor}-1) \] codewords.
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