Suppose the dose-response relation is quadratic in the bioassay experiment. Block designs which estimate preparation contrast (L(subscript p)), combined regression contrast (L1), parallelism cntrast (L1^1), quadratic curvature contrast (L2), and difference of quadratics contrast (L2^1) for the bioassay with full efficiency, are called Q-designs. In this paper, we present complete solutions of Q-designs for v≤16, where v is the total doses in the experiment. The method of construction relates to the existence of the quadratic trend-free block designs.