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This paper is on the channel training design for distributed space-time coding (DSTC) in multi-antenna relay networks. DSTC is shown to achieve full diversity in relay networks. To use DSTC, the receiver has to know both the channels between the relays and the receiver (Relay-Rx channels), and the channels between the transmitter and the relays (Tx-Relay channels). For the Relay-Rx channels, by sending pilot signals from the relays, the training problem can be solved using multi-input-multi-output (MIMO) training schemes. Given the knowledge of the Relay-Rx channels, to obtain estimations of the Tx-Relay channels at the receiver, DSTC is used. The linear minimum-mean-square-error (LMMSE) estimation at the receiver and the optimal pilot design that minimizes the estimation error are derived. We also investigate the requirement on the training time that can lead to full diversity in data transmission. An upper bound and a lower bound on the training time are provided. A novel training design whose training time length is adaptive to the quality of the Relay-Rx channels is also proposed. Simulations are exhibited to justify our analytical results and to show advantages of the proposed scheme over others.