Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs

TL;DR

This paper proves that extended perfect binary one-error-correcting codes can be uniquely reconstructed from their minimum distance graphs, establishing a one-to-one correspondence between the codes and their graphs and linking their automorphism groups.

Contribution

It provides a constructive proof for reconstructing these codes from their minimum distance graphs and shows the isomorphism of automorphism groups.

Findings

Codes can be uniquely reconstructed from their minimum distance graphs.

Automorphism groups of codes and their graphs are isomorphic.

Inequivalent codes have nonisomorphic minimum distance graphs.

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.