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Integrative design of communication mechanism and coordinated control law is an interesting and important problem for multi-agent networks. In this paper, we consider distributed coordination of discrete-time second-order multi-agent systems with partially measurable state and a limited communication data rate. A quantized-observer based encoding-decoding scheme is designed, which integrates the state observation with encoding/decoding. A distributed coordinated control law is proposed for each agent which is given in terms of the states of its encoder and decoders. It is shown that for a connected network, 2-bit quantizers suffice for the exponential asymptotic synchronization of the states of the agents. The selection of controller parameters and the performance limit are discussed. It is shown that the algebraic connectivity and the spectral radius of the Laplacian matrix of the communication graph play key roles in the closed-loop performance. The spectral radius of the Laplacian matrix is related to the selection of control gains, while the algebraic connectivity is related to the spectral radius of the closed-loop state matrix. Furthermore, it is shown that as the number of agents increases, the asymptotic convergence rate can be approximated as a function of the number of agents, the number of quantization levels (communication data rate) and the ratio of the algebraic connectivity to the spectral radius of the Laplacian matrix of the communication graph.