Hamming Code

CORE DEFINITION

In the field of telecommunications, the Hamming code, also known as the Hamming code, is a linear error-correcting code derived from the (7,4) Hamming code, invented by Richard Wesley Hamming in 1950. In contrast, simple parity check codes can only detect an odd number of errors and cannot correct them. The Hamming code is a perfect code, achieving the highest code rate among codes with the same block length and a minimum distance of 3. In mathematical terms, the Hamming code is a binary linear code. For all integers r ≥ 2, there exists a code with block length n = 2^r − 1 and k = 2^r − r − 1. Therefore, the code rate of the Hamming code is R = k / n = 1 − r / (2^r − 1), which is the highest for codes with minimum distance 3 and block length 2^r − 1. The parity check matrix of the Hamming code is constructed by listing all non-zero column vectors of length r.