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Consider the problem where I want to quantize a univariate standard Gaussian with two bits, and the objective is to minimize MSE reconstruction loss. According to these slides, the Lloyd algorithm and the entropy-constrained Lloyd algorithm give the optimal quantizer for fixed-length and variable-length coding respectively.

I would like to use fixed-length coding for computational efficiency, but the variable-length coding has a much smaller MSE (0.12 vs 0.09). Since both codings have the same average length, I wonder if I can approximate the variable-length coding by "aggregating" fixed-length coding. For example, maybe I can aggregate 1024 Gaussian samples together, and represent the vector in $\mathbb{R}^{1024}$ with 2048 bits. On average it's still 2 bits per sample, but the coding would have more flexibility, just like its variable-length counterpart.

How do I choose the partition and codebook for such a fixed-length coding?

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