Reconstructing extended perfect binary one-error-correcting codes from their minimum distance graphs

Authors:
Ivan Yu. Mogilnykh;Patric R. J. Östergård;Olli Pottonen;Faina I. Solov'eva
Affiliations:
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences and Novosibirsk State University, Novosibirsk, Russia;Department of Communications and Networking, Helsinki University of Technology, Finland;Department of Communications and Networking, Helsinki University of Technology, Finland;Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences and Novosibirsk State University, Novosibirsk, Russia
Venue:
IEEE Transactions on Information Theory
Year:
2009

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Abstract

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.