How to investigate the cells from the quantizer/encoder ${\rm round}_\mathbb{Z} \tilde{E}$ where $\tilde{E}$ is $m \times N$ on the set $X$ from before ($1 \geq |x_{(1)}| \geq 2|x_{(2)}| \geq \ldots \geq N|x_{(N)}|$)

In any case, need to cover our set X with the quantization cells.

Are the cells concentrated near the axes?

Is there any experiment I can run?

First study the signal on the boundary:

- Take random permutation $P$ and signs $\xi$
- Set $x(P(i)) = \xi(i)/i$

First question: Can I determine what cell it is in, and what the size is? $q = {\rm round}_\mathbb{Z}\tilde{E} x$. What other signals in $X$?

Next question: What properties must $\tilde{E}$ have for the cell to be small?

How to determine the set of all signals that map to this $q$?

The region of measurements that rounds to $q$ we know... $<e_i,x> = q_i \pm 0.5$ where $e_i$ are the entries of $\tilde{E}$. How about the inverse image of this region? Some notes:

- It's a convex region, and a linear transformation, so we only need to map the endpoints ($2^m$ of them), and take the convex hull.
- For any given measurement, the preimage under $\tilde{E}$ is a $N-m$ dimensional subspace (assume $\tilde{E}$ is full rank), which we then need to intersect with $X$.

This seems like quite the daunting task...

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