Constructing circuit codes by permuting initial sequences

Abstract: Two new constructions are presented for coils and snakes in the hypercube. Improvements are made on the best known results for snake-in-the-box coils of dimensions 9, 10 and 11, and for some other circuit codes of dimensions between 8 and 13. In the first construction, circuit codes are generated from permuted copies of an initial transition sequence; the multiple copies constrain the search, so that long codes can be found relatively efficiently. In the second construction, two lower-dimensional paths are joined together with only one or two changes in the highest dimension; this requires a search for a permutation of the second sequence to fit around the first. It is possible to investigate sequences of vertices of the hypercube, including circuit codes, by connecting the corresponding vertices in an extended graph related to the hypercube. As an example of this, invertible circuit codes are briefly discussed.