  Frameproof codes are used to preserve the security in the context of
coalition when fingerprinting digital data. Let $M_{c,l}(q)$ be the largest
cardinality of a $q$-ary $c$-frameproof code of length $l$ and
$R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}$. It has
been determined by Blackburn that $R_{c,l}=1$ when $l\equiv 1\ (\bmod\ c)$,
$R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}=5/3$. In this paper, we
give a recursive construction for $c$-frameproof codes of length $l$ with
respect to the alphabet size $q$. As applications of this construction, we
establish the existence results for $q$-ary $c$-frameproof codes of length
$c+2$ and size $\frac{c+2}{c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all
$q\equiv 4\pmod{6}$ when $c=3$. Furthermore, we show that $R_{c,c+2}=(c+2)/c$
meeting the upper bound given by Blackburn, for all integers $c$ such that
$c+1$ is a prime power.
