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In this paper, we consider the design of distributed space-time codes for wireless networks. Distributed space-time coding (DSTC) can be achieved through node cooperation to emulate multiple transmit antennas. We derive the distributed space-time codes design criteria for different scenarios based on the pairwise error probability (PEP) analysis. First, we consider the decode-and-forward (DAF) protocol and prove that space-time codes, designed to achieve full diversity and maximum coding gain in the MIMO channels, will achieve full diversity but not necessarily maximizing the coding gain if used with the DAF protocol. Next, we consider the amplify-and-forward (AAF) protocol and prove that a space-time code designed to achieve full diversity and maximum coding gain in MIMO channels will achieve the same if used with the AAF protocol.