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In this paper, we consider a decode-and-forward (DF) cooperation system consisting of two cooperative users in sending their information to a common destination, for which the distributed space-time coding (DSTC) is applied in an opportunistic manner, called opportunistic DSTC (O-DSTC), depending on whether the two users succeed in decoding each other's information or not. We propose two O-DSTC schemes for the full-duplex and half-duplex relaying scenarios, which are, respectively, referred to as the full-duplex and half-duplex-based O-DSTC. We evaluate the outage performance of the proposed O-DSTC as well as the conventional selective DF (S-DF) cooperation and fixed DSTC (F-DSTC) schemes. Numerical results show that the O-DSTC outperforms the conventional S-DF and F-DSTC schemes considering both full-duplex and half-duplex. In addition, we develop the diversity-multiplexing tradeoff (DMT) of the proposed O-DSTC, conventional S-DF and F-DSTC schemes by considering the two cooperative users with different data rates (also known as different multiplexing gains). We show that, for both the full-duplex and half-duplex modes, the proposed O-DSTC strictly outperforms the conventional S-DF and F-DSTC in terms of DMT. It is also shown that, in the full-duplex-based O-DSTC scheme, the diversity gain obtained by any of the two cooperative users not only depends on its own multiplexing gain, but also relates to its partner's multiplexing gain. By jointly considering the two users' DMT, the full-duplex-based O-DSTC scheme can achieve the optimal diversity gain when the two users are with the same multiplexing gain. For the half-duplex-based O-DSTC scheme, the DMT performance of the two users are independent of each other, which differs from the full-duplex-based O-DSTC scheme where mutual dependence exists between the cooperative users in terms of DMT.