  In the last two decades, several classes of codes are introduced to protect
the copyrighted digital data. They have important applications in the scenarios
like digital fingerprinting and broadcast encryption schemes. In this paper we
will discuss three important classes of such codes, namely, frameproof codes,
parent-identifying codes and traceability codes.
  Firstly, suppose $N(t)$ is the minimal integer such that there exists a
binary $t$-frameproof code of length $N$ with cardinality larger than $N$, we
prove that $N(t)\ge\frac{15+\sqrt{33}}{24} (t-2)^2$, which is a great
improvement of the previously known bound $N(t)\ge\binom{t+1}{2}$. Moreover, we
find that the determination of $N(t)$ is closely related to a conjecture of
Erd\H{o}s, Frankl and F\"uredi posed in the 1980's, which implies the
conjectured value $N(t)=t^2+o(t^2)$. Secondly, we derive a new upper bound for
parent-identifying codes, which is superior than all previously known bounds.
Thirdly, we present an upper bound for 3-traceability codes, which shows that a
$q$-ary 3-traceability code of length $N$ can have at most $cq^{\lceil
N/9\rceil}$ codewords, where $c$ is a constant only related to the code length
$N$. It is the first meaningful upper bound for 3-traceability codes and our
result supports a conjecture of Blackburn et al. posed in 2010.
