On the (n, t)-antipodal Gray codes

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 ≤ 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 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 ≤ k ≤ n - 3 when n ≥ 4, for t = 2^{n - k} when n ≥ 2 k with 1 ≤ k ≤ 3, for t = n when n = 2^k ≥ 2 (an alternative proof for Killian and Savage's result) ...etc.