Binary to Gray code (b XOR b shifted right) and back with a prefix XOR, and the table of all n-bit codes.
Procedure
How it works
In the Gray code (the reflected binary code, patented by Frank Gray in 1953) two consecutive numbers always differ in exactly one bit: 0, 1, 3, 2, 6, 7, 5, 4 in 3 bits is 000, 001, 011, 010, 110, 111, 101, 100. Ordinary binary does not have this property: going from 3 to 4 (011 to 100) flips all three bits.
The conversion is one XOR: $g = b \oplus (b \gg 1)$, so each Gray bit says whether a binary bit differs from its left neighbour. Going back, the top bit stays, and each further binary bit is the XOR of all Gray bits above and including it, a prefix XOR.
Gray codes are used where a reading may be taken while bits are changing, as in rotary and position encoders: a misread can only be off by one step. They also order the cells of Karnaugh maps and appear in genetic algorithms and error correction.
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