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For iterative detection and decoding (IDD) in multiple-input multiple-output (MIMO) systems, although the maximum a posteriori probability (MAP) detector is desirable in terms of performance, it is difficult to be employed due to its prohibitively high complexity as an exhaustive search is used. In this paper, a lattice reduction (LR)-based MIMO detection method is studied to achieve near MAP performance with reasonably low complexity for IDD. The a priori information (API), which is available from a soft-input soft-output (SISO) decoder, is taken into account to generate a list with a randomized successive interference cancellation (SIC) method. More specifically, a joint Gaussian distribution is used to convert the API into the LR domain and a modified sampling distribution, which was originally adopted for near optimal LR-based detection in non-IDD MIMO systems, is derived for random sampling to build a list of candidate vectors of high a posteriori probability (APP) with low complexity. It is shown that the IDD receiver with the proposed method outperforms those with the conventional LR-based methods, where no API is taken into account to build a list. Furthermore, the trade-off between performance and complexity is exploited with varying list length.