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In this paper, the error rate performance and diversity gain of amplify-and-forward (AF) wireless relay networks are studied. The distributed space-time block code (DSTBC) in a synchronous relay network with multiple relay nodes is considered. The distributed linear dispersion code is used at the relays. The symbol (or codeword) error rate is studied in terms of upper and lower bounds through rigorous derivations, and the diversity gain of the DSTBC is derived, for any number of relay nodes. As a corollary, a necessary and sufficient condition is obtained for achieving full cooperative diversity. Arbitrary coding matrices at the relays, path loss and general linear power allocation are considered.