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In this paper, we propose a novel distributed spacetime trellis code (DSTTC) structure, and analyze its error performance in both slow and quasi-slow Rayleigh fading channels. The protocol adopted is decode-and-forward (DAF) with a single relay between the source and destination. Both scenarios with perfect and imperfect decoding at the relay are investigated. For imperfect decoding at the relay node, we consider an equivalent one-hop link model for the source-relay-destination path, and use it to modify the maximum likelihood detection metric by taking into account the equivalent signal-to-noise ratio (SNR) of the link model. The upper bounds of pairwise error probability (PEP) are derived for slow and quasi-slow Rayleigh fading channels, and the DSTTC design criteria are formulated accordingly. Based on the proposed design criteria, new DSTTCs are constructed by computer search. Simulation results demonstrate the superiority of the designed codes.