Hamming codes

Hamming codes (by R. Hamming) – systematic block codes with parameters:

– code wordlength n = 2 ^r – 1;

– Quantity of information symbols k = 2 ^r – r – 1; (7.1)

– Number of additional symbols r = n – k;

– Minimum distance d _min = 3, r = 2, 3, 4.

Hamming codes –perfect codes which correct single errors.

By parameters choice r = 2, 3, 4 according to formulas (7.1) it is possible to set all known binary Hamming codes. For example, at r = 3 the parameters of a code (7, 4) will be the following:

– Length of the code word n = 3;

– Quantity of information symbols k = 4;

– Minimum distance d _min = 3;

– Code rate R _code = (2 ^r – r – 1)/(2 ^r – 1) = 4/7.

Generator and check matrixes of this code have been considered earlier, in section 4.1 (formulas (4.4) and (4.7)). As it has been noted earlier, this code allows to detect double errors also. Structures of encoder and syndrome decoder of Hamming code have been considered earlier in section 5.1 (figures 5.1, 5.2). According to the formula (3.5) transposed parity check matrix of this code looks like:

. (7.2)