Reconstructing Extended Perfect Binary One-Error-Correcting Codes From Their Minimum Distance Graphs
I. Yu. Mogilnykh, Patric R. J. Östergård, Olli Pottonen, F. I. Solov’eva · IEEE Transactions on Information Theory · 2009
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.