The structure of single-track Gray codes

Single-track Gray codes are cyclic Gray codes with codewords of length n, such that all the n tracks which correspond to the n distinct coordinates of the codewords are cyclic shifts of the first track. We investigate the structure of such binary codes and show that there is no such code with 2/sup n/ codewords when n is a power of 2. This implies that the known codes with 2/sup n/-2n codewords. when n is a power of 2, are optimal. This result is then generalized to codes over GF(p), where p is a prime. A subclass of single-track Gray codes, called single-track Gray codes with k-spaced heads, is also defined. All known systematic constructions for single-track Gray codes result in codes from this subclass. We investigate this class and show it has a strong connection with two classes of sequences, the full-order words and the full-order self-dual words. We present an iterative construction for binary single-track Gray codes which are asymptotically optimal if an infinite family of asymptotically optimal seed-codes exists. This construction is based on an effective way to generate a large set of distinct necklaces and a merging method for cyclic Gray codes based on necklaces representatives.