Gray code

Gray code is a binary numeral system where two successive values differ in only one bit.

For example, the sequence of Gray codes for 3-bit numbers is: 000, 001, 011, 010, 110, 111, 101, 100, so \(G(4) = 6\).

This code was invented by Frank Gray in 1953.

Finding Gray code

Let's look at the bits of number \(n\) and the bits of number \(G(n)\). Notice that \(i\)-th bit of \(G(n)\) equals 1 only when \(i\)-th bit of \(n\) equals 1 and \(i + 1\)-th bit equals 0 or the other way around (\(i\)-th bit equals 0 and \(i + 1\)-th bit equals 1). Thus, \(G(n) = n \oplus (n >> 1)\):

int g (int n) {
    return n ^ (n >> 1);
}

Finding inverse Gray code

Given Gray code \(g\), restore the original number \(n\).

We will move from the most significant bits to the least significant ones (the least significant bit has index 1 and the most significant bit has index \(k\)). The relation between the bits \(n_i\) of number \(n\) and the bits \(g_i\) of number \(g\):

\[ \begin{align} n_k &= g_k, \\ n_{k-1} &= g_{k-1} \oplus n_k = g_k \oplus g_{k-1}, \\ n_{k-2} &= g_{k-2} \oplus n_{k-1} = g_k \oplus g_{k-1} \oplus g_{k-2}, \\ n_{k-3} &= g_{k-3} \oplus n_{k-2} = g_k \oplus g_{k-1} \oplus g_{k-2} \oplus g_{k-3}, \\ \vdots \end{align} \]

The easiest way to write it in code is:

int rev_g (int g) {
  int n = 0;
  for (; g; g >>= 1)
    n ^= g;
  return n;
}

Practical applications

Gray codes have some useful applications, sometimes quite unexpected:

Practice Problems