�
6.895 Essential
Coding Theory
September
8,
2004
Lecture
1
Lecturer:
Madhu
Sudan
Scribe: Piotr
Mitros
1
Administrative
Madhu
Sudan
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do:
•
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up
for scribing – everyone
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even
listeners.
•
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•
Look at problem
set 1.
Part 1 due in
1 week.
2
Overview
of Class
Historical
overview.
The
field
started by a mathematician,
Hamming (1940s1950).
Hamming was
looking
at magnetic
storage
devices,
where data was
stored
in
32 bit chunks,
but bits
would
occasionally
be
ﬂipped.
He
was
investigating how
we could
do something better.
There are several
possible solutions:
Naive
solution:
Replicate
everything twice.
Will
correct a one bit error,
but for
every 3 bits,
we only
have 1
real
bit,
so the
rate
of
usage
is
only
1
.
3
Better
solution:
Divide
everything into
7 bit blocks.
In
each
block,
store 4 real
bits
such
that for any
one
bit ﬂipped,
we
can
figure
out which
bit was
ﬂipped.
Hamming did
this
with
the following encoding
process:
⎞
⎛
G
=
⎜
⎜
⎝
1
0
0
0
0
1
1
0
1
0
0
1
0
1
0
0
1
0
1
1
0
0
0
0
1
1
1
1
⎟
⎟
⎠
Encoding:
(
b
1
, b
2
, b
3
, b
4
)
−→
(
b
1
, b
2
, b
3
, b
4
)
G
·
Here,
the
multiplication
is
over
F
2
.
Claim:
If
a, b
∈
{
0
,
1
}
, a
=
b
then
a
·
G
and
b
·
G
differ
in
≥
3 coordinates.
This implies
that we
can
correct any one bit error,
since with
a one bit error,
we will
be one bit
away
from
the correct string,
and
at least 2 bits
away from the incorrect string.
Note we have not proven
this
claim
yet.
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 Spring '10
 Rejaei
 Electromagnet, Coding theory, Hamming Code, bit error, Hamming, bit Hamming code

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