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We study the asymptotic behavior of a cooperative scheme operating in amplify-and-forward (AF) mode when the number of relay terminals tends to infinity. We consider three time-division-multiple-access-based cooperation protocols which realize different degrees of broadcasting and receive collision. Optimal design rules for distributed space-time codes (STCs) are derived through the investigation of pairwise error probability expression for each considered protocol. The derived design rules reveal a Euclidean-distance-type performance metric, including some scaling terms due to the effects of AF operation mode and cooperation protocols as well as path loss/shadowing effects associated with the underlying relay links. Based on the derived criteria, an exhaustive computer code search is conducted to find out optimal distributed STCs. Our search results indicate the optimality of conventional designs, i.e., based on classical Euclidean distance, for relay numbers larger than two in a distributed scenario. For the special cases of one and two relays, we present novel distributed codes which are able to outperform their conventional counterparts and demonstrate robustness against distributed implementation.