Gray code, also called reflected binary code, is an ordering of binary numbers where two successive values differ in exactly one bit. This single-bit-change property reduces errors when reading mechanical or optical position sensors.

How is binary converted to Gray code?

The Gray code of a binary number n is computed as n XOR (n shifted right by one bit). The most significant bit is copied unchanged, and each subsequent Gray bit is the XOR of the current and previous binary bits.

How do you decode Gray code back to binary?

Decoding works by XOR-accumulating from the most significant bit: the first binary bit equals the first Gray bit, and each later binary bit is the previous binary bit XORed with the current Gray bit.

Why is Gray code used in rotary encoders?

Because only one bit changes between adjacent positions, a misread during a transition can only ever be off by one position rather than producing a large spurious jump, which dramatically improves reliability.

Is Gray code the same as standard binary?

No. Gray code and standard binary represent the same set of values but in a different bit pattern. For example decimal 3 is 011 in binary but 010 in Gray code. You must convert between them explicitly.

What is the Gray Code Converter?

Convert integers to reflected Gray code and decode Gray code back to standard binary. Gray code changes only one bit between consecutive values, making it ideal for rotary encoders and error reduction. It runs free in your browser on Gera Tools, with nothing uploaded.

Gray Code Converter — Gera Tools

Name: Gray Code Converter
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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Gray code in one minute

Gray code (reflected binary code) is a binary numbering system in which any two consecutive numbers differ in exactly one bit. That single-bit-change property is what makes it valuable for rotary encoders, Karnaugh maps, and digital communications where a transient misread should never cause a large jump in the decoded value.

How it works

To convert a binary number to Gray code, XOR the number with itself shifted one position to the right:

gray = n ^ (n >> 1)

The most significant bit is always copied unchanged; every other Gray bit is the XOR of the current binary bit and the bit immediately to its left.

Decoding reverses the process by accumulating XORs from the top bit down:

binary[0] = gray[0]
binary[i] = binary[i-1] ^ gray[i]

This converter applies these exact operations, so the encoded and decoded results are always bit-accurate for any non-negative integer within the safe integer range.

Example and tips

Decimal 4 is 100 in binary. Its Gray code is 100 ^ 010 = 110, which is decimal 6 when read as plain binary. Notice that the value 4 and its neighbour 5 (Gray 111) differ in only one bit — exactly the guarantee Gray code provides. When entering binary strings, leading zeros are preserved in the display so you can line up the bit columns for teaching or debugging.

Gray code conversion table (0–15)

Decimal	Binary	Gray code
0	0000	0000
1	0001	0001
2	0010	0011
3	0011	0010
4	0100	0110
5	0101	0111
6	0110	0101
7	0111	0100
8	1000	1100
9	1001	1101
10	1010	1111
11	1011	1110
12	1100	1010
13	1101	1011
14	1110	1001
15	1111	1000

Notice that adjacent rows always differ in exactly one bit position — that is the defining property of reflected binary Gray code.

Real-world applications

Rotary encoders. An absolute rotary encoder reads the angular position of a shaft using optical or magnetic sensors at fixed bit positions. If the shaft is between two position marks during a read, multiple bits might change simultaneously in standard binary, producing a transient reading far from either position. Gray code eliminates this: only one bit changes at any boundary, so the worst-case error is exactly one position step.

Karnaugh maps. In digital logic design, Karnaugh maps use Gray code ordering for their row and column labels so that adjacent cells differ in only one variable. This makes it easy to spot groupings of adjacent cells (implicants) that can be combined to simplify a Boolean expression.

Genetic algorithms and error correction. Some implementations of genetic algorithms encode numerical variables in Gray code to reduce the Hamming distance between adjacent values, which helps mutation operators make smaller incremental steps. It is also used in some error-correcting code designs where single-bit flip detection is important.