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# Appendices

## A.1 Hamming Code

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 = nm 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.

A.1 Generator matrix G is given from H as,

A.2 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,

A.3 It is noteworthy that all additional calculations are exclusive or (XOR) in this matrix calculation.

If Y is correctly converted,

A.4 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, ...

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