how to generalize gray coding

Question

I have a set of $2^n$ points in $d$-dimensional real space. This is my "constellation" $C \subset {\mathbb R}^d$. Let $B$ be the set of binary unordered tuples of length $n$. I want find a $1:1$ mapping $g : C \to B$ so that points in $C$ that are closest to each other (using euclidean distance) are mapped to points in $B$ that are closest to each other (using hamming distance). This is a very similar setting to "gray coding" hence the question title; although the problem might show up in different settings : "aligning two metrics" maybe ... or something along these lines.