An $n$ bit binary Gray code is a non-repeating sequence of the integers from $0$ to $2^n-1$ inclusive such that the binary representation of each number in the sequence differs by exactly one bit from the binary representation of the previous number: that is, the Hamming distance between consecutive elements is $1$ In addition, we also define a cyclic Gray code to be a Gray code where an extra condition is imposed: The last number in the sequence must differ by exactly one bit from the first number in the sequence.

For example, one $3$ bit cyclic Gray code is:

$$000_2$$ $$010_2$$ $$011_2$$ $$001_2$$ $$101_2$$ $$111_2$$ $$110_2$$ $$100_2$$

There is a one-to-one correspondence between all possible $n$ bit Gray codes and all possible Hamiltonian cycles on an $n$ dimensional hypercube. (To see why this is so, imagine assigning a binary number to each vertex of a hypercube where an edge joins each pair of vertices that differ by exactly one bit.)