# binary Gray code

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.)

Title binary Gray code BinaryGrayCode 2013-03-22 12:30:09 2013-03-22 12:30:09 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 68P30 msc 05C45 Gray code cyclic Gray code