We construct a randomized vector quantizer which has a smaller maximum error compared to all known lattice quantizers with the same entropy for dimensions 5, 6, ..., 48, and also has a smaller mean squared error compared to known lattice quantizers with the same entropy for dimensions 35, ..., 48, in the high resolution limit. Moreover, our randomized quantizer has a desirable property that the quantization error is always uniform over the ball and independent of the input. Our construction is based on applying rejection sampling on universal quantization, which allows us to shape the error distribution to be any continuous distribution, not only uniform distributions over basic cells of a lattice as in conventional dithered quantization. We also characterize the high SNR limit of one-shot channel simulation for any additive noise channel under a mild assumption (e.g., the AWGN channel), up to an additive constant of 1.45 bits.

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