Semi-definite relaxation (SDR) detector has been demonstrated to be successful in approaching maximum likelihood (ML) performance while the time complexity is only polynomial. We propose a new receiver jointly utilizing the forward error correction (FEC) code information in the SDR detection process. Strengthened by code constraints, the joint SDR detector substantially improves the overall receiver performance. For further performance boost, the ML-SDR detector is adapted to MAP-SDR detector by incorporating a priori information in the cost function. Under turbo principle, MAP-SDR detector takes in soft information from decoder and outputs extrinsic information with much improved reliability. We also propose a simplified SDR turbo receiver that solves only one SDR per codeword instead of solving multiple SDRs in the iterative turbo processing. This scheme significantly reduces the time complexity of SDR turbo receiver, while the error performance remains similar as before. In fact, our simplified scheme is generic and it can be applied to any list-based iterative receivers.