Hamming code ( in Chapter 7) is the most commonly used in computer applications. For given integer m, code length n is given by n = 2m − 1, while data length is given by k = n − m in Hamming code.
Typically, a check matrix H and a generator matrix G are utilized and should be defined in Hamming Code to detect and correct a data vector.
Generator matrix G is given from H as,
All column vector in matrices H and G must be independent and must not be null vector (all elements are ‘0’s).
When the data bits are x = [1 1 0 1], the codeword can be calculated by multiplying x and G,
It is noteworthy that all additional calculations are exclusive or (XOR) in this matrix calculation.
If Y is correctly converted,
If not, the result of the above calculation, syndrome, gives the same vector as the one of the column vectors of the check matrix H where one bit error is introduced. When two-bit errors take place, the calculated results indicate the wrong column vector. In that case, the following check matrix in the extended Hamming code is utilised, ...