Convert Octal to Gray Code

G = B XOR (B >> 1) – bit-by-bit XOR of binary with itself shifted right by one. Each Gray bit equals the binary bit at the same position XORed with the next-higher binary bit. For the leftmost (most-significant) Gray bit, the “next higher” bit is implicitly 0, so it equals the most-significant binary bit unchanged. This tool uses BigInt operators so arbitrary-size values work exactly.

Cumulative XOR: start with the Gray code as-is, then XOR each successive shift of the Gray code right by one position back into it: B = G XOR (G >> 1) XOR (G >> 2) XOR …. The tool implements this as a loop until G >> shift reaches zero. Each iteration recovers one binary bit; for 32-bit values, that’s at most 32 iterations.

That’s the defining property – it’s why Gray code exists. In regular binary, going from 7 (0111) to 8 (1000) flips all 4 bits at once. In a rotary encoder, those 4 bits change at slightly different times due to mechanical imperfection, so the reader might briefly see 0000, 1111, or any wrong intermediate value. Gray code ensures only one bit ever changes between adjacent positions, so even with imperfect timing the reading is either the old or the new value – never a spurious intermediate.

The exact algorithm. Most tutorials say “Gray = binary XOR (binary right-shifted)” without showing it bit by bit. The panel breaks it down: for each bit position, which binary bit XOR which prior binary bit produces which Gray bit. After working through the table, the formula G[i] = B[i] XOR B[i+1] becomes intuitive.

Yes. Standard Gray code is also called “reflected binary code” because of how you can construct it: start with [0, 1], reflect the list and prepend 0 to the first half and 1 to the second half: [00, 01, 11, 10]. Reflect again: [000, 001, 011, 010, 110, 111, 101, 100] – that’s 3-bit Gray. The “reflection” property gives the one-bit-change guarantee.

Rotary encoders (every robot arm joint sensor), shaft position sensors in industrial control, analog-to-digital converter outputs (some flash ADC topologies use Gray internally), Karnaugh maps in digital logic design (the row/column order is Gray to make adjacency obvious), and some genetic algorithms (mutations in Gray code change one bit, exploring the search space more smoothly).

No. Leading zeros never change between consecutive Gray codes (they don’t represent active bits), so padding with zeros is safe. 00000111 and 00000101 still differ in exactly one bit. This is why the tool happily pads to 8/16/32/64-bit without warning – the Gray property is preserved.

Verification. If you suspect the Gray conversion is wrong, you can check the intermediate binary representation against a decimal-to-binary calculator. Most BCD/Gray confusion stems from mixing up which representation you’re looking at – having binary, Gray, and decimal side-by-side eliminates the ambiguity.

Unlimited (BigInt). A 100-digit decimal becomes a ~333-bit Gray code, computed exactly. The result table cell will be wide; horizontal scroll keeps it usable. Practical use is usually under 64 bits.

No. All BigInt arithmetic (XOR, shifts, base conversions) happens in your browser. Open DevTools → Network and confirm zero requests fire after the page loads. Safe for proprietary encoder values or competitive-programming inputs.