Given $m$ points in $\mathbb{R}^N$, the Johnson-Lindenstrauss lemma guarantees the existence of a linear operator $\mathbb{R}^N\rightarrow\mathbb{R}^n$ that nearly preserves pairwise distances between the points. Here, we can take $n=\Omega(\log(m)/\varepsilon^2)$, where $\varepsilon$ is the level of distortion.

Is there a similar result for points in a vector space over a finite field, e.g. $\mathbb{F}_2^N$? I assume a result of this form would be in terms of Hamming distance.