The structure of single-track Gray codes

Single-track Gray codes are cyclic Gray codes with codewords of length n, s uch 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 str ucture of such binary codes and show that there is no such code with 2(n) c odewords when n is a power of 2, This implies that the known codes with 2(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-t rack Gray codes, called single-track Gray codes with k-spaced heads, is als o defined. All known systematic constructions for single-track Gray codes r esult in codes from this subclass. We investigate this class and show it ha s a strong connection with two classes of sequences, the full-order words a nd the full-order self-dual words. We present an iterative construction for binary single-track Gray codes which are asymptotically optimal if an infi nite 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 representativ es.