Frameproof codes are a class of secure codes that were originally introduced in the pioneering work of Boneh and Shaw in the context of digital fingerprinting. They can be used to enhance the security and credibility of digital content. Let $M_{c,l}(q)$ denote the largest cardinality of a $q$-ary $c$-frameproof code with length $l$. Based on an intriguing observation that relates $M_{c,l}(q)$ to the renowned Erd\H{o}s Matching Conjecture in extremal set theory, in 2003, Blackburn posed an open problem on the precise value of the limit $R_{c,l}=\lim_{q\rightarrow\infty}\frac{M_{c,l}(q)}{q^{\lceil l/c \rceil}}$. By combining several ideas from the probabilistic method, we present a lower bound for $M_{c,l}(q)$, which, together with an upper bound of Blackburn, completely determines $R_{c,l}$ for {\it all} fixed $c,l$, and resolves the above open problem in the full generality. We also present an improved upper bound for $M_{c,l}(q)$.

翻译：防陷害码是一类安全码，最初由Boneh和Shaw在数字指纹领域的开创性工作中引入。它们可用于增强数字内容的安全性和可信度。设$M_{c,l}(q)$表示长度为$l$的$q$元$c$-防陷害码的最大基数。基于一个有趣的现象——将$M_{c,l}(q)$与极值集合论中著名的Erdős匹配猜想联系起来，Blackburn于2003年提出了一个关于极限$R_{c,l}=\lim_{q\rightarrow\infty}\frac{M_{c,l}(q)}{q^{\lceil l/c \rceil}}$精确值的开放问题。通过结合概率方法中的若干思路，我们给出了$M_{c,l}(q)$的一个下界，该下界与Blackburn的一个上界相结合，完全确定了对于{\it 所有}固定$c,l$的$R_{c,l}$值，从而在完全一般性上解决了上述开放问题。我们还给出了$M_{c,l}(q)$的一个改进上界。