HammingKodes

Hamming codes are a class of linear error-correcting codes that can detect up to two-bit errors or correct single-bit errors. They were invented by Richard Hamming in 1950. Hamming codes are widely used in memory systems, such as RAM, and in data transmission to ensure data integrity.

The basic principle behind Hamming codes is the addition of parity bits to the data bits. These parity bits are calculated based on specific combinations of data bits. When the data is transmitted or stored, the parity bits are sent along with the data. Upon reception or retrieval, the parity bits are recalculated and compared with the received parity bits. Any discrepancies can indicate the presence and location of errors.

A Hamming code is typically denoted by H(k, n), where k is the number of data bits and n is the total number of bits (data bits plus parity bits). The number of parity bits, p, is determined by the inequality 2^p >= n + 1. For example, the Hamming(7,4) code uses 4 data bits and 3 parity bits, for a total of 7 bits. This code can detect all single and double-bit errors and correct all single-bit errors.

The process of encoding involves placing data bits in specific positions and calculating parity bits based on the positions of the data bits they cover. Decoding involves checking the parity of received bits. If all parities match, there is no error. If there is a mismatch, the sum of the positions of the mismatched parity bits indicates the location of the error, allowing for its correction. Hamming codes are a fundamental concept in error detection and correction techniques.