On the (n,t)-antipodal Gray codes

  • Authors:

  • Gerard J. Chang;Sen-Peng Eu;Chung-Heng Yeh

  • Affiliations:

  • Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan and National Centre for ...;Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 81148, Taiwan;Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

Quantified Score

Hi-index 5.23

Visualization

Abstract

An n-bit Gray code is a circular listing of all 2^nn-bit binary strings in which consecutive strings differ at exactly one bit. For n@?t@?2^n^-^1, an (n,t)-antipodal Gray code is a Gray code in which the complement of any string appears t steps away from the string, clockwise or counterclockwise. Killian and Savage proved that an (n,n)-antipodal Gray code exists when n is a power of 2 or n=3, and does not exist for n=6 or odd n3. Motivated by these results, we prove that for odd n=3, an (n,t)-antipodal Gray code exists if and only if t=2^n^-^1-1. For even n, we establish two recursive constructions for (n,t) codes from smaller (n^',t^'). Consequently, various (n,t)-antipodal Gray codes are found for even n's. Examples are for t=2^n^-^1-2^k with k odd and 1@?k@?n-3 when n=4, for t=2^n^-^k when n=2k with 1@?k@?3, for t=n when n=2^k=2 (an alternative proof for Killian and Savage's result) ...etc.