On the (n,t)-antipodal Gray codes
An n-bit Gray code is a circular listing of all 2(n) n-bit binary strings in which consecutive strings differ at exactly one bit. For 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 = 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  = 4, for t = 2(n-k) when n >= 2k with 1  = 2 (an alternative proof for Killian and Savage's result) ...etc. (C) 2006 Elsevier B.V. All rights reserved.
gray code antipodal hamiltonian cycle n-cube complement period