Wu et al. (IEEE Signal Process Mag 21:15–27, 2004) showed that fingerprinting system can be designed according to different criteria: catching one, catching many and catching all. For the case of “catching many”, binary multimedia identifiable parent property codes ( t -MIPPCs) were introduced to construct fingerprints resistant to the averaging collusion attack on multimedia contents. In this paper, we focus on such a case, and introduce the binary strongly multimedia identifiable parent property code ( t -SMIPPC) whose tracing algorithm is more efficient than that of a binary t -MIPPC. Then a concatenation construction for binary t -SMIPPCs from q -ary t -SMIPPCs is provided. Moreover, we establish a probabilistic lower bound on q -ary t -SMIPPCs, whose code rate is asymptotically as good as that of q -ary t -MIPPCs. Finally, several infinite series of optimal q -ary t -SMIPPCs of length 2 with t = 2 , 3 are derived from the relationships among t -SMIPPCs and other fingerprinting codes, such as t -MIPPCs and t ¯ -separable codes.