Encoding Process

c= uG

Decoding Process

s= rH^T

If some output values of the encoding and decoding multiplication is over than 1, the value should be changed into remainder on division of itself by 2.  (Ex: (4 2 1 ) => (0 0 1))

s(Syndrome) detect error position. We can know the single-error-bit position in the table below.

Fig. Decoding table for Hamming code(7,4)

If s is [0 0 0] and r= [0 1 0 0 0 1 1], it supposes that there is no error and transmitted data is [0 0 1 1]. If s is [0 1 0] and r = [1 0 1 0 0 1 0], it supposes that 2^th bit is corrupted, so r should be changed to [1 1 1 0 0 1 0], and transmitted data is lower 4bits, [0 0 1 0].