Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let M-c,M-l(q) be the largest cardinality of a q-ary c-frameproof code of length l and R-c,R-l = lim(q ->infinity) M-c,M-l(q)/q([l/c]). It has been determined by Blackburn that R-c,R-l = 1 when l equivalent to 1 (mod c), R-c,R-l = 2 when c = 2 and l is even, and R-3,R-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 c+2/c(q-1)(2) + 1 for all odd when c = 2 and for all q equivalent to 4 (mod 6) when c = 3. Furthermore, we show that R-c,R-c+2 = (c + 2)/c meeting the upper bound given by Blackburn, for all integers c such that c + 1 is a prime power.