In this paper, we design the precoding matrices at the base station side and the combining matrices at the user terminal side for initial downlink synchronization in millimeter wave massive multiple-input multiple-output systems. First, we demonstrate two basic requirements for the precoding and combining matrices, including that all the entries therein should have constant amplitude under the implementation architecture constraint, and the average transmission power over the total K time slots taking for synchronization should be constant for any spatial direction. Then, we derive the optimal synchronization detector based on generalized likelihood ratio test. By utilizing this detector, we analyze the effect of the precoding and combining matrices to the missed detection probability and the false alarm probability, respectively, and present the corresponding conditions that should be satisfied. It is shown that, both of the precoding and combining matrices should guarantee the perfect omnidirectional coverage at each time slot, i.e., the average transmission power at each time slot is constant for any spatial direction, which is more strict than the second basic requirement mentioned above. We also show that such omnidirectional precoding matrices and omnidirectional combining matrices exist only when both of the number of transmit streams and the number of receive streams are equal to or greater than two. In this case, we propose to utilize Golay complementary pairs and Golay-Hadamard matrices to design the precoding and combining matrices. Simulation results verify the effectiveness of the propose approach.