Error correction Hamming(7,4)
a bit error on bit 5 causes bad parity in red , green circles
the diagram right shows bit error (shown in blue text) , bad parity created (shown in red text) in red , green circles. bit error can detected computing parity of red, green, , blue circles. if bad parity detected data bit overlaps bad parity circles bit error. in above example, red , green circles have bad parity bit corresponding intersection of red , green not blue indicates errored bit.
now,
z
=
h
r
=
(
1
0
1
0
1
0
1
0
1
1
0
0
1
1
0
0
0
1
1
1
1
)
(
0
1
1
0
1
1
1
)
=
(
3
4
3
)
=
(
1
0
1
)
{\displaystyle \mathbf {z} =\mathbf {hr} ={\begin{pmatrix}1&0&1&0&1&0&1\\0&1&1&0&0&1&1\\0&0&0&1&1&1&1\\\end{pmatrix}}{\begin{pmatrix}0\\1\\1\\0\\1\\1\\1\end{pmatrix}}={\begin{pmatrix}3\\4\\3\end{pmatrix}}={\begin{pmatrix}1\\0\\1\end{pmatrix}}}
which corresponds fifth column of h. furthermore, general algorithm used (see hamming code#general algorithm) intentional in construction syndrome of 101 corresponds binary value of 5, indicates fifth bit corrupted. thus, error has been detected in bit 5, , can corrected (simply flip or negate value):
r
corrected
=
(
0
1
1
0
1
¯
1
1
)
=
(
0
1
1
0
0
1
1
)
{\displaystyle \mathbf {r} _{\text{corrected}}={\begin{pmatrix}0\\1\\1\\0\\{\overline {1}}\\1\\1\end{pmatrix}}={\begin{pmatrix}0\\1\\1\\0\\0\\1\\1\end{pmatrix}}}
this corrected received value indeed, now, matches transmitted value x above.
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