TY - JOUR

T1 - On the (n, t)-antipodal Gray codes

AU - Chang, Gerard J.

AU - Eu, Sen Peng

AU - Yeh, Chung Heng

N1 - Funding Information:
The authors thank the referee for many constructive suggestions. First author was partially supported by the National Science Council under grant NSC 95-2221-E-002-125-MY3. Second author was partially supported by the National Science Council under grant NSC 95-2115-M-390-006-MY3 and TJ&MY Foundation.

PY - 2007/4/20

Y1 - 2007/4/20

N2 - An n-bit Gray code is a circular listing of all 2n n-bit binary strings in which consecutive strings differ at exactly one bit. For n ≤ t ≤ 2n - 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 n > 3. Motivated by these results, we prove that for odd n ≥ 3, an (n, t)-antipodal Gray code exists if and only if t = 2n - 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 = 2n - 1 - 2k with k odd and 1 ≤ k ≤ n - 3 when n ≥ 4, for t = 2n - k when n ≥ 2 k with 1 ≤ k ≤ 3, for t = n when n = 2k ≥ 2 (an alternative proof for Killian and Savage's result) ...etc.

AB - An n-bit Gray code is a circular listing of all 2n n-bit binary strings in which consecutive strings differ at exactly one bit. For n ≤ t ≤ 2n - 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 n > 3. Motivated by these results, we prove that for odd n ≥ 3, an (n, t)-antipodal Gray code exists if and only if t = 2n - 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 = 2n - 1 - 2k with k odd and 1 ≤ k ≤ n - 3 when n ≥ 4, for t = 2n - k when n ≥ 2 k with 1 ≤ k ≤ 3, for t = n when n = 2k ≥ 2 (an alternative proof for Killian and Savage's result) ...etc.

KW - Antipodal

KW - Complement

KW - Gray code

KW - Hamiltonian cycle

KW - Period

KW - n-cube

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U2 - 10.1016/j.tcs.2006.12.005

DO - 10.1016/j.tcs.2006.12.005

M3 - Article

AN - SCOPUS:33947432006

SN - 0304-3975

VL - 374

SP - 82

EP - 90

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1-3

ER -