The structure of single-track Gray codes

Moshe Schwartz, Tuvi Etzion

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

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 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.

Original languageEnglish
Pages (from-to)2383-2396
Number of pages14
JournalIEEE Transactions on Information Theory
Volume45
Issue number7
DOIs
StatePublished - 1 Dec 1999
Externally publishedYes

Keywords

  • Cyclic Gray codes
  • Feedback shift-register
  • Linear complexity
  • Necklaces
  • Self-dual sequences
  • Single-track codes

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

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