Error-correcting codes on the towers of Hanoi graphs Original Research Article

A perfect one-error-correcting code on a graph is a subset of the vertices so that no two vertices in the subset are adjacent and each vertex not in the subset is adjacent to exactly one vertex in the subset. We show that the Towers of Hanoi puzzle defines an infinite family of graphs, and that each such graph supports a perfect one-error-correcting code. We show that these codes are essentially unique. Our characterization of the codewords as those ternary strings with an even number of 1ʹs and an even number of 2ʹs, makes generation and decoding computationally easy. In particular, decoding can be carried out by a two-pass finite state machine. We also show that determining if a graph can support a perfect one-error-correcting code is an NP-complete problem.