This paper deals with the reachable set synthesis problem for discrete-time periodic piecewise systems (PPSs) subject to multiple packet dropouts and bounded-peak disturbances. Firstly, the multiple packet dropout scenario is formulated by using two mutually independent binary variables satisfying a Bernoulli distribution. Then, a novel Lyapunov function related to the control matrix information is constructed to develop the periodic state feedback controller in the observer framework. The proposed controller can ensure the stability and reachability of the closed-loop system in the mean-square sense. For the sake of calculations, the singular value decomposition (SVD) condition is employed to design the observer and controller, thereby transforming into a convex programming problem with linear matrix inequality (LMI) constraints. Moreover, an index optimizing the bounding region of the desirable reachable set is given via resorting to the ellipsoid technique, and their results are compared. Finally, numerical examples are given to validate the effectiveness of the proposed theoretical results.