Antipodal Gray codes Original Research Article

An n-bit Gray code is a circular listing of the 2n 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.