In this paper, downlink non-orthogonal multiple access (NOMA) with receivers performing single-user decoding i.e., without successive interference cancellation (SIC) is studied. Using lattice partitions, we generalize the scheme recently proposed by Shieh and Huang  to general n-dimensional constellations carved from lattices. The achievable rates of the proposed scheme without SIC and the gap to the capacity region are investigated. Design examples based on lattice partition chains in Z2, A2, and D4 are studied. Numerical and simulation results are provided, which demonstrate advantages of the proposed scheme over the one in  and any orthogonal multiple access scheme.