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In this paper, the design of distributed space-time (ST) codes for wireless relay networks is considered. Distributed ST coding (DSTC) can be achieved through node cooperation to emulate a multiple-antenna transmitter. First, the decode-and- forward (DAF) protocol, in which each relay node decodes the symbols received from the source node before retransmission, is considered. An ST code designed to achieve full diversity and maximum coding gain over multiple-input-multiple-output (MIMO) channels is proven to achieve full diversity but not necessarily maximize the coding gain if used with the DAF protocol. Next, the amplify-and-forward (AAF) protocol is considered; each relay node can only perform simple operations, such as linear transformation of the received signal and amplification of the signal before retransmission. An ST code designed to achieve full diversity and maximum coding gain over MIMO channels is proven to achieve full diversity and maximum coding gain if used with the AAF protocol. Next, the design of DSTC that can mitigate the relay node synchronization errors is considered. Most of the previous works on cooperative transmission assume perfect synchronization between the relay nodes, which means that the relays' timings, carrier frequencies, and propagation delays are identical. Perfect synchronization is difficult to achieve among randomly located relay nodes. To simplify the synchronization in the network, a diagonal structure is imposed on the ST code used. The diagonal structure of the code bypasses the perfect synchronization problem by allowing only one relay node to transmit at any time slot. Hence, it is not necessary to synchronize simultaneous "in-phase" transmissions of randomly located relay nodes, which greatly simplifies the synchronization among the relay nodes. The code design criterion for distributed ST codes based on the diagonal structure is derived. This paper shows that the code design criterion maximizes the minimum product d- - istance.