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In this paper, we study the Space Time Block Code (STBC) called Matrix D (MD) that enjoys low-complexity detection for broadcast transmission. Such a transmission uses a powerful channel code like a turbo code. To produce Log-Likelihood Ratio (LLR) for the channel decoder, we propose a soft Maximum Likelihood (ML) detector for MD STBC while keeping the same low-complexity order as the original one. Then, we optimize the MD STBC for low Signal to Noise Ratio (SNR) under the trace criterion. The performance of the concatenation of a turbo code with optimized MD is given. A gain of almost 0.2 dB is obtained between optimized MD and original MD. However, the proposed code loses its full-diversity as its determinant is equal to 0. Then, a tradeoff angle is chosen and a low-complexity detection full-rate full-diversity STBC is designed for turbo coded broadcast transmission which performs well for both low and high SNR.