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In this paper, we study linear precoding for multiple-input multiple-output (MIMO) multiple access channels (MAC) with discrete-constellation inputs. We derive the constellation-constrained capacity region for the MIMO MAC with an arbitrary number of users. Due to the non-concavity of the objective function, we obtain the necessary conditions for the weighted sum rate (WSR) maximization problem through Karush-Kuhn-Tucker (KKT) analysis. To find the optimal precoding matrices, we propose an iterative algorithm utilizing alternating optimization strategy and gradient descent update. Numerical results show that when inputs are digital modulated signals and the signal-to-noise ratio (SNR) is in the medium range, our proposed algorithm offers significantly higher sum rate than non-precoding and the traditional method which maximizes Gaussian-input sum capacity. Furthermore, the bit error rate (BER) results of a low-density parity-check (LDPC) coded system also indicate that the system with the proposed linear precoder achieves significant gains over other methods.