Gray Code

The Gray code, also called reflected binary, is a binary code having the property of modifying only one bit when a number is increased (or decreased) by one unit.

This property can have several interesting practical applications, and the gray code appears in Baudot code, in Hanoi towers resolution, or position encoders.

To transform binary into reflected binary (Gray code), the algorithm consists in calculating the exclusive OR (XOR) between the binary value and itself but shifted by a row to the right (the last bit is deleted).

The algorithm implementation in computers languages is done in one line and uses binary operators xor and shift: function bin2gray(n) return n ^ (n >> 1)

An algorithm for converting an integer to Gray code (binary) uses successive divisions by powers of 2 and looks at the parity of the rounded quotient. (Thanks G. Plousos)

Another conversion method, more visual, is described by this image (Thanks G. Plousos) :

Gray code conversion can be done bit by bit. Given a number $ G = {g_0,g_1,\dots,g_n} $ with $ g_i $ each of its bits, then $ B = {b_0,b_1,\dots,b_n} $ is calculated as: $$ b_0 = g_0 \\ b_1 = g_0 \oplus g_1 \\ b_2 = g_0 \oplus g_1 \oplus g_2 \\ b_n = g_0 \oplus g_1 \oplus \dots \oplus g_n $$

In gray code, the most significant bit ($ g_0 $, often on the left) is always the same as the binary one ($ b_0 $).

The implementation of the conversion calculation also uses the xor and shift binary operators: function gray2bin(n1) {
n2 = n1;
while (n1 >>= 1) n2 ^= n1;
return n2;
}

Gray Code allows you to count in binary by modifying a single bit to go from one number to the next. Here are the first 16 values in 4-bit gray code:

0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, 1100, 1101, 1111, 1110, 1010, 1011, 1001, 1000

The first equivalent decimal values are: 0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 24, 25, 27, 26, 30, 31 , 29, 28, 20, 21, 23, 22, 18, 19, 17, 16, etc. here

Gray code is modified only one bit at once when incrementing, which simplifies calculations and speed them up in some cases.

Gray code is difficult to distinguish from other binary code.

The color gray (also written grey) is a clue.

The Gray code is protected by a patent from 1953.