Hamming code 7 4

Hamming code 7 4 is the first code in the class of Hamming codes. Error syndromes, corresponding to single-error patterns, are written as binary words:
that encode a position of nonzero bit in z , as presented in Table
i 1
1000000
001
2
0100000
010
3
0010000
011
4
0001000
100
5
0000100
101
6
0000010
110
7
0000001
111
8
0000000
000
As already observed (Problem 5.2.2), syndrome of the i -th single error is the i -th column of the parity check matrix. Then, parity-check matrix:
As can be seen, the Hamming code is the nonsystematic one. In such code, it is convenient to select check-bit positions such, that each check-bit is designated by information bits only. Thus, columns of parity-check matrix that contain only one nonzero entry designate check-bits positions. For the (7,4) Hamming code, these positions are: . Then, the codeword:
where, parity-check bits can be designated from equations:
.
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