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This paper presents an optimal dynamic quantizer synthesis for controlling linear systems with the discrete-valued input. The quantizers considered here are in the form of a difference equation, for which we find quantizer parameters such that the system composed of a given linear plant and the quantizer is an optimal approximation of the linear plant in terms of the input-output relation. First, the performance of the dynamic quantizers is analyzed, and then a closed form expression of the performance is derived. Next, in spite of the nonconvexity of the design problem, a globally optimal solution is provided via linear programming. Finally, the validity of the proposed method is demonstrated by numerical simulations.