It is known [4] that for any numbers n = 2m- 1, m ≥ 4 and r, such that n - log(n + 1) ≤ r ≤ n there exists a perfect binary code of length n and rank r. We show that there exists a propelinear such code of length n, excluding, may be, n = r = 63, n = 127, r ϵ {126,127} and n = r = 2047.

The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect binary one-error-correcting code from its minimum distance graph is presented. Consequently, inequivalent such codes have nonisomorphic minimum distance graphs. Moreover, it is shown that the automorphism group of a minimum distance graph is isomorphic to that of the corresponding code.