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In this paper, an Irregular Distributed Space-Time (Ir-DST) coding scheme is studied in the context of a twin-relay aided network in which the successive relaying protocol is employed. A tight upperbound of the successive relaying aided network's capacity is given. The distributed codes at the source and relays are jointly designed with the aid of Extrinsic Information Transfer (EXIT) charts for the sake of high-integrity operation at Signal-to-Noise Ratios (SNRs) close to the corresponding network's capacity. Finally, it is shown that our proposed Ir-DST coding scheme is capable of near-capacity cooperative communications in the successive relaying aided network, which is an explicit benefit of our joint source-and-relay mode design.