  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 $M_{c,l}(q)$.
