Arslan showed that computing all-pairs Hamming distances is easily reducible to arithmetic 0–1 matrix multiplication (IPL 2018). We provide a reverse, linear-time reduction of arithmetic 0–1 matrix multiplication to computing all-pairs distances in a Hamming space. On the other hand, we present a fast randomized algorithm for approximate all-pairs distances in a Hamming space. By combining it with our reduction, we obtain also a fast randomized algorithm for approximate 0–1 matrix multiplication. Finally, we present an output-sensitive randomized algorithm for a minimum spanning tree of a set of points in a generalized Hamming space, the lower is the cost of the minimum spanning tree the faster is our algorithm.(A preliminary version of this article appeared in arXiv.org.)