Dirty paper coding (DPC) is applied to linear multi-input multi-output (MIMO) broadcast channels with additive white Gaussian noise and one message per receiver. The method decomposes each receiver channel into parallel scalar channels with known interference, then applies modulo operators, amplitude-shift keying (ASK), and probabilistic shaping. The achievable rate tuples include all points inside the capacity region by choosing truncated Gaussian shaping, large ASK alphabets, and large modulo intervals. Simulations with short polar codes show significant rate gains from DPC compared to conventional linear precoding, while maintaining similar encoder and decoder complexities.