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Linear-assignment Gel'fand-Pinsker coding (LA-GPC) is a coding technique for channels with interference known only at the transmitter, where the known interference is treated as side-information (SI). As a special case of the LA-GPC, dirty paper coding has been shown to be able to achieve the optimal interference-free rate for SI channels with perfect channel state information at the transmitter (CSIT). In the cases where only the channel distribution information at the transmitter (CDIT) is available, LA-GPC also has good (sometimes optimal) performance in a variety of fast and slow fading SI channels. In this letter, we design filters in the nested lattice based coding to make it achieve the same rate performance as the LA-GPC in multiple-input multiple-output (MIMO) channels. Compared with the random Gaussian codebooks used in previous works, our resultant coding schemes have algebraic structures and can be implemented in practical systems. Simulations in slow-fading channels are provided, and near interference-free error performance is obtained. The proposed coding schemes can serve as the fundamental building blocks to achieve the promised rate performance of MIMO Gaussian broadcast channels with CDIT or perfect CSIT. [ABSTRACT FROM AUTHOR]