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We investigate the design of linear transmit precoding for multiple-input multiple-output (MIMO) broadcast channels (BC) with finite alphabet input signals. We derive an explicit expression for the achievable rate region of the MIMO BC with discrete constellation inputs, which is generally applicable to cases involving arbitrary user number and arbitrary antenna configurations. For the case where all the users employ the same modulation scheme, we further present a weighted sum rate upper-bound of the MIMO BC with identical transmit precoding matrices. The resulting bound demonstrates a serious performance loss due to multi-user interference for MIMO BC with finite alphabet inputs in high signal-to-noise ratio (SNR) region, which motivates the use of simple precoding to combat this multiuser interference. Based on a constrained optimization problem formulation, we apply the Karush-Kuhn-Tucker analysis to derive necessary conditions for MIMO BC precoders to maximize the weighted sum-rate. We then propose an iterative gradient descent algorithm with backtracking line search to optimize the linear precoders for each user. Numerical results illustrate that our proposed algorithm provides significant gains over other conventional precoding schemes including the traditional iterative water-filling (WF) design for the Gaussian input assumption.