Totally balanced and exponentially balanced Gray codes

The method of Robinson and Cohn to construct balanced and totally balanced Gray codes is discussed, as well as the extended version of this method by Bhat and Savage. We introduce a slight generalization of their construction which enables us to prove a long standing conjecture of Wagner and West about the existence of Gray codes having a specific spectrum of transition counts, i.e., all transition counts are powers of 2 and the exponents of these powers differ at most 1. Such a Gray code can be considered as generalization of a totally balanced Gray code when the length of the codewords is not a 2-power.