Antipodal Gray codes

An n-bit Gray code is a circular listing of the 2 n-bit strings so that successive strings differ only in one bit position. An n-bit antipodal Gray code has the additional property that the complement of any string appears exactly n steps away in the list. The problem of determining for which values of n antipodal Gray codes can exist was posed by Hunter Snevily, who showed them to be possible for n = 1, 2, 3, and 4. In this paper, we show they are not possible for odd n > 3 or for n = 6. However, we provide a recursive construction to prove existence when n is a power of 2. The question remains open for any even n > 6 which is not a power of 2.