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Hamming Code in Computer Network

Last Updated : 14 May, 2025

Hamming code is an error-correcting code used to ensure data accuracy during transmission or storage. Hamming code detects and corrects the errors that can occur when the data is moved or stored from the sender to the receiver. This simple and effective method helps improve the reliability of communication systems and digital storage. It adds extra bits to the original data, allowing the system to detect and correct single-bit errors. It is a technique developed by Richard Hamming in the 1950s.

Redundant bits are extra binary bits that are generated and added to the information-carrying bits of data transfer to ensure that no bits were lost during the data transfer. The number of redundant bits can be calculated using the following formula:

 2r ≥ m + r + 1 

where,

• m is the number of bits in input data
• r is the number of redundant bits.

Suppose the number of data bits is 7, then the number of redundant bits can be calculated as

= 24 ≥ 7 + 4 + 1 

Thus, the number of redundant bits is 4.

A parity bit is a bit appended to a data of binary bits to ensure that the total number of 1’s in the data is even or odd. Parity bits are used for error detection. There are two types of parity bits:

• Even Parity Bit: In the case of even parity, for a given set of bits, the number of 1’s are counted. If that count is odd, the parity bit value is set to 1, making the total count of occurrences of 1’s an even number. If the total number of 1’s in a given set of bits is already even, the parity bit's value is 0.
• Odd Parity Bit: In the case of odd parity, for a given set of bits, the number of 1’s are counted. If that count is even, the parity bit value is set to 1, making the total count of occurrences of 1’s an odd number. If the total number of 1’s in a given set of bits is already odd, the parity bit's value is 0.

Hamming Code is simply the use of extra parity bits to allow the identification of an error.

Step 1: Write the bit positions starting from 1 in binary form (1, 10, 11, 100, etc).

Step 2: All the bit positions that are a power of 2 are marked as parity bits (1, 2, 4, 8, etc).

Step 3: All the other bit positions are marked as data bits.

Step 4: Each data bit is included in a unique set of parity bits, as determined its bit position in binary form:

• Parity bit 1 covers all the bits positions whose binary representation includes a 1 in the least significant position (1, 3, 5, 7, 9, 11, etc).
• Parity bit 2 covers all the bits positions whose binary representation includes a 1 in the second position from the least significant bit (2, 3, 6, 7, 10, 11, etc).
• Parity bit 4 covers all the bits positions whose binary representation includes a 1 in the third position from the least significant bit (4–7, 12–15, 20–23, etc).
• Parity bit 8 covers all the bits positions whose binary representation includes a 1 in the fourth position from the least significant bit bits (8–15, 24–31, 40–47, etc).
• In general, each parity bit covers all bits where the bitwise AND of the parity position and the bit position is non-zero.

Step 5: Since we check for even parity set a parity bit to 1 if the total number of ones in the positions it checks is odd. Set a parity bit to 0 if the total number of ones in the positions it checks is even.

A redundancy bits are placed at positions that correspond to the power of 2. As in the above example:

• The number of data bits = 7
• The number of redundant bits = 4
• The total number of bits = 7+4=11
• The redundant bits are placed at positions corresponding to power of 2 that is 1, 2, 4, and 8

• Suppose the data to be transmitted is 1011001 from sender to receiver, the bits will be placed as follows: 

• R1 bit is calculated using parity check at all the bits positions whose binary representation includes a 1 in the least significant position. R1: bits 1, 3, 5, 7, 9, 11 

• To find the redundant bit R1, we check for even parity. Since the total number of 1’s in all the bit positions corresponding to R1 is an even number. So, the value of R1 (parity bit’s value) = 0.

• R2 bit is calculated using parity check at all the bits positions whose binary representation includes a 1 in the second position from the least significant bit. R2: bits 2,3,6,7,10,11 

• To find the redundant bit R2, we check for even parity. Since the total number of 1’s in all the bit positions corresponding to R2 is odd the value of R2(parity bit’s value)=1
• R4 bit is calculated using parity check at all the bits positions whose binary representation includes a 1 in the third position from the least significant bit. R4: bits 4, 5, 6, 7 

•  To find the redundant bit R4, we check for even parity. Since the total number of 1’s in all the bit positions corresponding to R4 is odd so the value of R4(parity bit’s value) = 1
• R8 bit is calculated using parity check at all the bits positions whose binary representation includes a 1 in the fourth position from the least significant bit. R8: bit 8,9,10,11  

• To find the redundant bit R8, we check for even parity. Since the total number of 1’s in all the bit positions corresponding to R8 is an even number the value of R8(parity bit’s value)=0. Thus, the data transferred is:

Suppose in the above example the 6th bit is changed from 0 to 1 during data transmission, then it gives new parity values in the binary number: 

For all the parity bits we will check the number of 1's in their respective bit positions.

• For R1: bits 1, 3, 5, 7, 9, 11. We can see that the number of 1's in these bit positions are 4 and that's even so we get a 0 for this.
• For R2: bits 2,3,6,7,10,11 . We can see that the number of 1's in these bit positions are 5 and that's odd so we get a 1 for this.
• For R4: bits 4, 5, 6, 7 . We can see that the number of 1's in these bit positions are 3 and that's odd so we get a 1 for this.
• For R8: bit 8,9,10,11 . We can see that the number of 1's in these bit positions are 2 and that's even so we get a 0 for this.
• The bits give the binary number 0110 whose decimal representation is 6. Thus, bit 6 contains an error. To correct the error the 6th bit is changed from 1 to 0.

• Error Detection and Correction: Hamming code is designed to detect and correct single-bit errors that may occur during the transmission of data. This ensures that the recipient receives the same data that was transmitted by the sender.
• Redundancy: Hamming code uses redundant bits to add additional information to the data being transmitted. This redundancy allows the recipient to detect and correct errors that may have occurred during transmission.
• Efficiency: Hamming code is a relatively simple and efficient error-correction technique that does not require a lot of computational resources. This makes it ideal for use in low-power and low-bandwidth communication networks.
• Widely Used: Hamming code is a widely used error-correction technique and is used in a variety of applications, including telecommunications, computer networks, and data storage systems.
• Single Error Correction: Hamming code is capable of correcting a single-bit error, which makes it ideal for use in applications where errors are likely to occur due to external factors such as electromagnetic interference.
• Limited Multiple Error Correction: Hamming code can only correct a limited number of multiple errors. In applications where multiple errors are likely to occur, more advanced error-correction techniques may be required.

For Implementation you can refer this article.

Which one of the following choices gives the correct values of x and y ?

(A) x is 0 and y is 0

(B) x is 0 and y is 1

(C) x is 1 and y is 0

(D) x is 1 and y is 1

Answer: (A)

We will first insert our codeword according to hamming code d₈d₇d₆d₅c₄d₄d₃d₂c₃d₁c₂c_1,

Now, calculating hamming code according to first parity bit C1: d₇d₅d₄d₂d₁c_1. 1x0010, To make number of 1 even , for this x must be 0.

Similarly, lets calculate for y , we will start from c₈ and make its even=>110xy here x is already 0 , so y should be 0.

So the value of x is 0 and y is 0.

For more details you can refer GATE | GATE CS 2021 | Set 1 | Question 39 published quiz.

• Hamming code can detect and correct single-bit errors, enhancing data reliability during transmission and storage.
• It adds a minimal number of redundant bits to the original data, maintaining a good balance between data integrity and overhead. The algorithm for generating and checking Hamming code is straightforward and can be easily implemented in both hardware and software.
• By detecting and correcting errors, Hamming code ensures that the received data is accurate, reducing the chances of data corruption.
• Hamming code is widely used in various fields such as computer memory (RAM), data storage devices, and communication systems.
• Compared to more complex error correction codes, Hamming code provides a cost-effective solution for applications where single-bit error correction is sufficient.

• Hamming code can only correct single-bit errors. It is unable to correct multiple-bit errors, which limits its effectiveness in environments with high error rates.
• While it can detect single-bit and some two-bit errors, Hamming code cannot detect all multiple-bit errors. This reduces its reliability in certain applications.
• Although it uses fewer redundant bits compared to some other error correction methods, the addition of these bits still increases the overall data size, which can be a drawback in bandwidth-constrained environments.
• Implementing Hamming code requires additional hardware or software resources for error detection and correction, which can be a limitation in resource-constrained systems.

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