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We consider t-write codes for write-once memories with n cells that can store multiple levels. Assuming an underlying lattice-based construction and using the continuous approximation, we derive upper bounds on the worst-case sum-rate optimal and fixed-rate optimal n-cell t-write write-regions for the asymptotic case of continuous levels. These are achieved using hyperbolic shaping regions that have a gain of 1 bit/cell over cubic shaping regions. Motivated by these hyperbolic write-regions, we discuss construction and encoding of codebooks for cells with discrete support. We present a polynomial-time algorithm to assign messages to the codebooks and show that it achieves the optimal sum-rate for any given codebook when n = 2. Using this approach, we construct codes that achieve high sum-rate. We describe an alternative formulation of the message assignment problem for n≥ 3, a problem which remains open.