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Hamming Distance
vectors in
(or more generally
)
Hamming Distance, the distance between
and
Ex
d(
,
- d(
,
- d(
,
- d(
,
- d(
) = 0
(Codes) is C= set of codewords
We define the Hamming distance for C as
here we took
so that
we did not get d(x,x)=0.
We will use hamming distance to find the codeword closest to the
decoded word and to correct an error in our encoded word. This is
called ``maximum likelihood decoding". We won't be able to to this
for the ``parity bit" codes but the ``Hamming codes" will let us
fix one error (and detect up to three) in a received word.
Ex Minimum Distance for
C=
Because C is a linear code, then:
Minimum distance is 2.
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Frederick Leitner
2004-05-12