Reconstructing Extended Perfect Binary One-Error-Correcting Codes from   Their Minimum Distance Graphs
  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. 
Ivan Yu. Mogilnykh, Patric R. J. \"Osterg{\aa}rd, Olli Pottonen, Faina   I. Solov'eva