Hamming code

Hamming code is a technique used to detect and correct errors in data transmission. It is a type of error-correcting code that adds extra parity bits to the original data bits, which allows errors to be detected and corrected.

The basic steps of using Hamming code for error detection and correction are as follows:

1. Determine the number of parity bits required: The number of parity bits required is the smallest number that satisfies the equation 2^r >= m + r + 1, where m is the number of data bits and r is the number of parity bits.
2. Place the data bits in the appropriate positions: The data bits are placed in the positions that are powers of 2, starting with position 1. The remaining positions are reserved for the parity bits.
3. Calculate the parity bits: For each parity bit, determine the positions of the data bits that cover that bit, and calculate the parity for those bits. The parity bit is set to 1 if the total number of 1s is odd, and 0 if the total number of 1s is even.
4. Transmit the message: The original data bits and the calculated parity bits are transmitted together.
5. Check for errors: At the receiving end, the data bits and the parity bits are checked for errors. If an error is detected, the receiver uses the parity bits to correct the error.

Example:

Suppose we want to transmit the 4-bit data word 1010 using a Hamming code with 3 parity bits.

1. Determine the number of parity bits required: 2^r >= m + r + 1   2³ >= 4 + 3 + 1   r = 3
2. Place the data bits in the appropriate positions: 1 0 1 0 P 0 P 0 P P 1 0 P P P XThe data bits are placed in positions 1, 2, 4, and 8. The parity bits are placed in positions 3, 5, and 6. The position X is unused.
3. Calculate the parity bits: Position 3: Parity for bits 1, 2, 4 = 1 Position 5: Parity for bits 1, 4, 8 = 0 Position 6: Parity for bits 2, 4, 8 = 1The calculated parity bits are 101.
4. Transmit the message: The transmitted message is 1011010.
5. Check for errors: Suppose the received message is 1001010. The receiver calculates the parity bits as follows:Position 3: Parity for bits 1, 2, 4, 7 = 1 (error) Position 5: Parity for bits 1, 4, 8 = 0 Position 6: Parity for bits 2, 4, 8 = 1

   The calculated parity bits are 011.

   The receiver can use the parity bits to correct the error in position 3. The corrected message is 1011010.