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In this paper, we jointly consider statistical precoding and feedback of bit allocation (BA) for multiple-input–multiple-ouput (MIMO) systems over correlated channels. The proposed system will be termed a BA system. We assume that the statistics of the channel is known to the transmitter and a reverse link of very low rate is available so that the receiver can send back the quantized BA. Based on the statistics of the channel, we derive the statistical precoder so that bounds of the error rate averaged over the random correlated channel are minimized. Due to statistical precoding, the distribution of the BA is highly nonuniform. Treating BA as a vector signal, we quantize it using vector quantization (VQ), which is known to be particularly useful for quantizing signals with nonuniform distributions. As the distribution of BA is taken into consideration in the codebook design, a good tradeoff between performance and feedback rate can be achieved. Simulations show that the combination of statistical precoding and VQ-based quantization for BA leads to good performance with a small number of feedback bits. [ABSTRACT FROM PUBLISHER]