It is well known that you can communicate a discrete signal via a continuous but noisy one (see the noisy channel coding theorem). My question is, can you do the opposite in the case of gaussian noise?
Given any $r \in \mathbb R_+$, does there exist $n \in \mathbb N$ (the number of discrete levels) and independent random functions $e : [0, r) \to \{0, 1, \dots, n\}$ (the encoding function) and $d : \{0, 1, \dots, n\} \to \mathbb R$ (the decoding function) such that:
$$d(e(x)) \sim \mathcal{N}(r, 1)$$
for all $x \in [0, r)$?
In other words, can we communicate a real number in the range from $0$ to $r$ using a discrete channel, such that the noise is the standard normal distribution.
I am guessing the minimum possible $n$ to be approximately proportional to $r$.