A result of Johnson and Lindenstrauss shows that a set of $n$ points in high dimensional Euclidean space can be mapped into an $O(\frac{\log n}{\epsilon^2})$- dimensional Euclidean space such that the distance between any two points changes by only a factor of $(1\pm \epsilon).$
I am looking for reference for a similar result for Hamming space- i.e. a set of $n$ binary vectors can be mapped into a very low dimensional space such that the hamming distance between any pair of vectors is approximately preserved.