Joel Adams' Gray Code Converter Applet

The gray code number system has the property that each number i's successor and predecessor numbers differ from i in just one bit. To illustrate, the applet below lets you enter a decimal number and find the corresponding gray code value, or vice versa:

By entering a sequence of consecutive decimal values (e.g., 0, 1, 2, 3, 4, 5, 6, 7), you can determine the corresponding consecutive gray-code numbers. Note that for any two consecutive decimal numbers, the base-2 representation of the corresponding gray code values differ in exactly 1 bit.

These gray code numbers can be used to map an alternative topology like a ring onto a hypercube. For example, a degree 3 hypercube has 8 nodes. To map a ring onto those 8 nodes, we visit them in order of the gray code numbers corresponding to the consecutive decimal values 0 through 7:

  000, 001, 011, 010, 110, 111, 101, 100

Since an N-dimensional hypercube consists of two N-1-dimensional hypercubes, we can assign each node in an N-dimensional hypercube a gray-code number using this methodology:

• If (N == 1), assign 0 to one node and 1 to the other.
• If (N > 1), a node's gray-code number is its degree-N-1 hypercube gray-code number, with
  • a 0 prepended to that number in all of the nodes of one of the degree-N-1 hypercubes; and
  • a 1 prepended to that number in all of the nodes of the other degree-N-1 hypercube.

In an N-dimensional hypercube, each of the low-order N bits in a node's gray-code number can be associated with a different dimension of the hypercube, and each dimension corresponds to a different link for that node. For a hypercube, node whose decimal number is i, the bit that is different between gray-code(i) and gray-code(i+1) tells us which link we should follow from i to visit the next node in the ring.