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Showing content from https://www.geeksforgeeks.org/dsa/gray-to-binary-and-binary-to-gray-conversion/ below:

Gray to Binary and Binary to Gray conversion

Last Updated : 23 Jul, 2025

Try it on GfG Practice

Binary Number is the default way to store numbers, but in many applications, binary numbers are difficult to use and a variety of binary numbers is needed. This is where Gray codes are very useful.

Gray code has a property that two successive numbers differ in only one bit because of this property gray code does the cycling through various states with minimal effort and is used in K-maps, error correction, communication, etc.

How to Convert Binary To Gray and Vice Versa?

Binary : 0011
Gray : 0010

Binary : 01001

Gray : 01101

Binary to Gray conversion :

The Most Significant Bit (MSB) of the gray code is always equal to the MSB of the given binary code.
Other bits of the output gray code can be obtained by XORing binary code bit at that index and previous index.

Binary code to gray code conversion

Gray to binary conversion :

The Most Significant Bit (MSB) of the binary code is always equal to the MSB of the given gray code.
Other bits of the output binary code can be obtained by checking the gray code bit at that index. If the current gray code bit is 0, then copy the previous binary code bit, else copy the invert of the previous binary code bit.

Gray code to binary code conversion

Below is the implementation of the above approach:

C++


 // C++ program for Binary To Gray
// and Gray to Binary conversion
#include <bits/stdc++.h>
using namespace std;

// Function to find xor of two 
// bits represented as character.
char xorChar(char a, char b) { 
    return (a == b) ? '0' : '1'; 
}

// Function to flip a bit
// represented as character.
char flip(char c) { 
    return (c == '0') ? '1' : '0'; 
}

// function to convert binary string
// to gray string
string binToGrey(string binary) {
    string gray = "";

    // MSB of gray code is same as binary code
    gray += binary[0];

    // Compute remaining bits, next bit is computed by
    // doing XOR of previous and current in Binary
    for (int i = 1; i < binary.length(); i++) {
        
        // Concatenate XOR of previous bit
        // with current bit
        gray += xorChar(binary[i - 1], binary[i]);
    }

    return gray;
}

// function to convert gray code string
// to binary string
string greyToBin(string gray) {
    string binary = "";

    // MSB of binary code is same as gray code
    binary += gray[0];

    // Compute remaining bits
    for (int i = 1; i < gray.length(); i++) {
        
        // If current bit is 0, concatenate
        // previous bit
        if (gray[i] == '0')
            binary += binary[i - 1];

        // Else, concatenate invert of
        // previous bit
        else
            binary += flip(binary[i - 1]);
    }

    return binary;
}

int main() {
    string binary = "01001";
    cout << binToGrey(binary) << endl;

    string gray = "01101";
    cout << greyToBin(gray) << endl;
    return 0;
}

Java


 // Java program for Binary To Gray
// and Gray to Binary conversion

class GfG {

    // Function to find xor of two 
    // bits represented as character.
    static char xorChar(char a, char b) { 
        return (a == b) ? '0' : '1'; 
    }

    // Function to flip a bit
    // represented as character.
    static char flip(char c) { 
        return (c == '0') ? '1' : '0'; 
    }

    // function to convert binary string
    // to gray string
    static String binToGrey(String binary) {
        String gray = "";

        // MSB of gray code is same as binary code
        gray += binary.charAt(0);

        // Compute remaining bits, next bit is computed by
        // doing XOR of previous and current in Binary
        for (int i = 1; i < binary.length(); i++) {
            
            // Concatenate XOR of previous bit
            // with current bit
            gray += xorChar(binary.charAt(i - 1), binary.charAt(i));
        }

        return gray;
    }

    // function to convert gray code string
    // to binary string
    static String greyToBin(String gray) {
        String binary = "";

        // MSB of binary code is same as gray code
        binary += gray.charAt(0);

        // Compute remaining bits
        for (int i = 1; i < gray.length(); i++) {
            
            // If current bit is 0, concatenate
            // previous bit
            if (gray.charAt(i) == '0')
                binary += binary.charAt(i - 1);

            // Else, concatenate invert of
            // previous bit
            else
                binary += flip(binary.charAt(i - 1));
        }

        return binary;
    }

    public static void main(String[] args) {
        String binary = "01001";
        System.out.println(binToGrey(binary));

        String gray = "01101";
        System.out.println(greyToBin(gray));
    }
}

Python


 # Python program for Binary To Gray
# and Gray to Binary conversion

# Function to find xor of two 
# bits represented as character.
def xorChar(a, b): 
    return '0' if a == b else '1'

# Function to flip a bit
# represented as character.
def flip(c): 
    return '1' if c == '0' else '0'

# function to convert binary string
# to gray string
def binToGrey(binary):
    gray = ""

    # MSB of gray code is same as binary code
    gray += binary[0]

    # Compute remaining bits, next bit is computed by
    # doing XOR of previous and current in Binary
    for i in range(1, len(binary)):
        
        # Concatenate XOR of previous bit
        # with current bit
        gray += xorChar(binary[i - 1], binary[i])

    return gray

# function to convert gray code string
# to binary string
def greyToBin(gray):
    binary = ""

    # MSB of binary code is same as gray code
    binary += gray[0]

    # Compute remaining bits
    for i in range(1, len(gray)):
        
        # If current bit is 0, concatenate
        # previous bit
        if gray[i] == '0':
            binary += binary[i - 1]

        # Else, concatenate invert of
        # previous bit
        else:
            binary += flip(binary[i - 1])

    return binary

if __name__ == "__main__":
    binary = "01001"
    print(binToGrey(binary))

    gray = "01101"
    print(greyToBin(gray))


 // C# program for Binary To Gray
// and Gray to Binary conversion
using System;
using System.Text;

class GfG {

    // Function to find XOR of two 
    // bits represented as characters
    static char xorChar(char a, char b) { 
        return (a == b) ? '0' : '1'; 
    }

    // Function to flip a bit 
    // represented as a character
    static char flip(char c) { 
        return (c == '0') ? '1' : '0'; 
    }

    // Function to convert binary 
    // string to gray string
    static string binToGrey(string binary) {
        StringBuilder gray = new StringBuilder();

        // MSB of gray code is 
        // same as binary code
        gray.Append(binary[0]);

        // Compute remaining bits
        for (int i = 1; i < binary.Length; i++) {
            gray.Append(xorChar(binary[i - 1], binary[i]));
        }

        return gray.ToString();
    }

    // Function to convert gray code 
    // string to binary string
    static string greyToBin(string gray) {
        StringBuilder binary = new StringBuilder();

        // MSB of binary code is same as gray code
        binary.Append(gray[0]);

        // Compute remaining bits
        for (int i = 1; i < gray.Length; i++) {
            if (gray[i] == '0')
                binary.Append(binary[i - 1]); 
            else
                binary.Append(flip(binary[i - 1])); 
        }

        return binary.ToString();
    }

    static void Main() {
        string binary = "01001";
        Console.WriteLine(binToGrey(binary));

        string gray = "01101";
        Console.WriteLine(greyToBin(gray));
    }
}

JavaScript


 // JavaScript program for Binary To Gray
// and Gray to Binary conversion

// Function to find xor of two 
// bits represented as character.
function xorChar(a, b) { 
    return (a === b) ? '0' : '1'; 
}

// Function to flip a bit
// represented as character.
function flip(c) { 
    return (c === '0') ? '1' : '0'; 
}

// function to convert binary string
// to gray string
function binToGrey(binary) {
    let gray = "";

    // MSB of gray code is same as binary code
    gray += binary[0];

    // Compute remaining bits, next bit is computed by
    // doing XOR of previous and current in Binary
    for (let i = 1; i < binary.length; i++) {
        
        // Concatenate XOR of previous bit
        // with current bit
        gray += xorChar(binary[i - 1], binary[i]);
    }

    return gray;
}

// function to convert gray code string
// to binary string
function greyToBin(gray) {
    let binary = "";

    // MSB of binary code is same as gray code
    binary += gray[0];

    // Compute remaining bits
    for (let i = 1; i < gray.length; i++) {
        
        // If current bit is 0, concatenate
        // previous bit
        if (gray[i] === '0')
            binary += binary[i - 1];

        // Else, concatenate invert of
        // previous bit
        else
            binary += flip(binary[i - 1]);
    }

    return binary;
}

let binary = "01001";
console.log(binToGrey(binary));

let gray = "01101";
console.log(greyToBin(gray));

Time Complexity: O(n), where n is length of the binary string.
Auxiliary Space: O(n)

If the binary code and gray code is given in integer format, then we can use bitwise operators to convert the codes. Below is the implementation using bitwise operators:

C++


 // C++ program for Binary To Gray
// and Gray to Binary conversion
#include <bits/stdc++.h>
using namespace std;

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

int greyToBin(int n) {
    int res = n;
    while (n > 0) {
        n >>= 1;
        res ^= n;
    }
    return res;
}

int main() {
    int binary = 3;
    cout << binToGrey(binary) << endl;
  
  	int gray = 2;
  	cout << greyToBin(gray) << endl;
    return 0;
}

Java


 // Java program for Binary To Gray
// and Gray to Binary conversion

class GfG {

    // Function to convert binary to gray
    static int binToGrey(int n) { 
        return n ^ (n >> 1); 
    }

    // Function to convert gray to binary
    static int greyToBin(int n) {
        int res = n;
        while (n > 0) {
            n >>= 1;
            res ^= n;
        }
        return res;
    }

    public static void main(String[] args) {
        int binary = 3;
        System.out.println(binToGrey(binary));
        
        int gray = 2;
        System.out.println(greyToBin(gray));
    }
}

Python


 # Python program for Binary To Gray
# and Gray to Binary conversion

# Function to convert binary to gray
def binToGrey(n): 
    return n ^ (n >> 1)

# Function to convert gray to binary
def greyToBin(n):
    res = n
    while n > 0:
        n >>= 1
        res ^= n
    return res

if __name__ == "__main__":
    binary = 3
    print(binToGrey(binary))
  
    gray = 2
    print(greyToBin(gray))


 // C# program for Binary To Gray
// and Gray to Binary conversion

using System;

class GfG {

    // Function to convert binary to gray
    static int binToGrey(int n) { 
        return n ^ (n >> 1); 
    }

    // Function to convert gray to binary
    static int greyToBin(int n) {
        int res = n;
        while (n > 0) {
            n >>= 1;
            res ^= n;
        }
        return res;
    }

    static void Main() {
        int binary = 3;
        Console.WriteLine(binToGrey(binary));

        int gray = 2;
        Console.WriteLine(greyToBin(gray));
    }
}

JavaScript


 // JavaScript program for Binary To Gray
// and Gray to Binary conversion

// Function to convert binary to gray
function binToGrey(n) { 
    return n ^ (n >> 1); 
}

// Function to convert gray to binary
function greyToBin(n) {
    let res = n;
    while (n > 0) {
        n >>= 1;
        res ^= n;
    }
    return res;
}

let binary = 3;
console.log(binToGrey(binary));

let gray = 2;
console.log(greyToBin(gray));

Time Complexity:

For binary to gray conversion: O(1)
For gray to binary conversion: O(log(n)), as n is a decimal number and number of bits is approximately log(n).

Auxiliary Space: O(1)

Related Article:

Generate n-bit Gray Codes

Binary To Gray Code Equivalent | DSA Problem

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