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For the separation of the signals in the vector broadcast channel (BC), some information about the channel state is necessary at the transmitter. In many cases, this channel state information (CSI) must be fed back from the receivers to the transmitter. We jointly design the channel estimators and the quantizers at the receivers together with the precoder at the transmitter based on a precoder-centric criterion, i.e., the minimization of a mean square error (MSE) metric appropriate for the precoder design. This is in contrast to our previous works, where the quantizer design was based on a CSI MSE metric, i.e., based on the minimization of the MSE between the true channel and the channel recovered by the transmitter using a feedback channel. Interestingly, the estimators resulting from this joint formulation are independent of the used codebook. The codebook entries are the employed precoders. Therefore, each receiver feeds back the index of a set of precoders and the intersection of the sets gives the appropriate precoder. Since the quantizers of the different receivers have to work separately, the metric for the computation of the partition cells cannot be expressed as a simple squared error depending on the quantizer output. The proposed system based on a joint optimization clearly outperforms previous designs with separate optimization of feedback and precoding.