Technical Report CS0937

Title: The Structure of Single-Track Gray Codes
Authors: Moshe Schwartz, Tuvi Etzion
Abstract: 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 $2^n$ codewords 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-track Gray codes, called single-track Gray codes with evenly 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-period necklaces and the full-period self-dual necklaces. 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 nonequivalent necklaces and a merging method for cyclic Gray codes based on necklaces representatives.
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