In this paper, we study upper bounds on the minimum length of frameproof codes introduced by Boneh and Shaw to protect copyrighted materials. A $q$-ary $(k,n)$-frameproof code of length $t$ is a $t \times n$ matrix having entries in $\{0,1,\ldots, q-1\}$ and with the property that for any column $\mathbf{c}$ and any other $k$ columns, there exists a row where the symbols of the $k$ columns are all different from the corresponding symbol (in the same row) of the column $\mathbf{c}$. In this paper, we show the existence of $q$-ary $(k,n)$-frameproof codes of length $t = O(\frac{k^2}{q} \log n)$ for $q \leq k$, using the Lov\'asz Local Lemma, and of length $t = O(\frac{k}{\log(q/k)}\log(n/k))$ for $q > k$ using the expurgation method. Remarkably, for the practical case of $q \leq k$ our findings give codes whose length almost matches the lower bound $\Omega(\frac{k^2}{q\log k} \log n)$ on the length of any $q$-ary $(k,n)$-frameproof code and, more importantly, allow us to derive an algorithm of complexity $O(t n^2)$ for the construction of such codes.

翻译：本文研究了Boneh和Shaw为保护版权材料而引入的防框架代码的最小长度上界。一个长度为t的q元(k,n)-防框架代码是一个t×n矩阵，其条目取自{0,1,…,q-1}，并具有如下性质：对于任意列c和任意其他k列，存在某一行，使得该行中这k列的符号均与列c的对应符号（位于同一行）不同。本文利用Lovász局部引理证明了当q≤k时，存在长度为t=O((k²/q) log n)的q元(k,n)-防框架代码；当q>k时，通过删截法证明了存在长度为t=O((k/log(q/k)) log(n/k))的代码。值得注意的是，对于q≤k这一实际场景，我们的结果给出了长度几乎匹配任何q元(k,n)-防框架代码长度下界Ω((k²/(q log k)) log n)的代码，更重要的是，由此可推导出复杂度为O(tn²)的此类代码构造算法。