3.13 Hamming Code

3.13 Hamming Code
The first class of binary linear block code discovered by R. W.
Hamming (1915 – 1998)
R. W. Hamming, “Error detecting and error correcting codes”,
Bell System Technical Journal, vol. 29, pp. 147 – 160, 1950.
For any positive integer m ≥ 3, there exists a Hamming code with
the following parameters:
block code length
m
n = 2 - 1
message length
k = 2
m
- 1 -
m
minimum Hamming distance d min
= 3
error-correction capability t = 1
Example:(7, 4) Hamming code (Example 3.1 & 3.5)
G
⎡1
⎢
0
= ⎢
⎢0
⎢
⎣0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
1
0
0
1
1
1
1⎤
1
⎥
⎥
0⎥
⎥
1⎦
⎡1
H =
⎢
0
⎢
⎢⎣
1
1
1
1
1
1
0
0
1
1
1
0
0
0
1
0
0⎤
0
⎥
⎥
1⎥⎦

For a Hamming code of length 2 m - 1 , its parity-check matrix is
a matrix whose columns consist of the entire set of the non-zero
binary m-tuples.
3.14 Golay Code
The (23,12) Gloay code is the only known multiple-errorcorrecting
binary perfect code, which is capable of correcting 3
or fewer random errors in a block of 23 digits, d min
= 7 .
(Discovered by Golay in 1949).
The (23, 12) Gloay code is either generated by
g +
2 4 5 6 10 11
1(
x)
= 1 + x + x + x + x + x x
or by
g +
5 6 7 9 11
2
( x)
= 1 + x + x + x + x + x x
Both g 1
(x)
and g 2
(x)
are factors of x 23 + 1
23
and + 1 = ( 1 + x)
g ( x)
g ( x)
x
1 2

Golay’s paper was published in 1949.
M. J. E. Golay, “Notes on digital coding”, proc. IRE, 37, pp. 567,
June 1949.
Golay codes have been frequent application in the US space
program, most notably with the Voyager I and II spacecraft,
providing (1979 – 81) clear color pictures of Jupiter and Saturn.
3.15 Reed-Muller Code (RM Code)
Reed-Muller codes were discovered by Muller in 1954, and
shortly thereafter a better algebraic representation was provided
by Reed along with an elegant decoding algorithm.
The first-order RM code of length 32 wasusedin1969forthe
error-control system of the Mariner and Viking deep-space
probes of Mars.

The original r-th order RM codes were binary and noncyclic, and
formed a special subclass of Euclidean geometry (EG) codes.
(Details can also be found in Lin / Costello Book, chap. 8)
The r-th order binary RM code has the following parameters:
m
n = 2
k
⎛m⎞
⎛m⎞
⎛m⎞
= 1 + ⎜ ⎟ + ⎜ ⎟ + L+
⎜ ⎟
⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ r ⎠
n - k
= 2
m
- k
=
m
∑ − r − 1
i=
0
⎛
⎜
⎝
m
i
⎞
⎟
⎠
m-r
d
min
= 2
Where r

Let v0
be a vector whose
m
2 components are all 1s. Then v 0
is called an identity vector.
Let
v L be the rows of a matrix with all possible
1
,v2
, , vm
m-tuples as columns. We call them the basis vectors.
The r-th order RM code is formed by using the basis vectors
v L and all of their vector products r or fewer at a time.
1
,v2
, , vm
The generator matrix of the r-th order RM code is defined as
⎡ v0
⎢
v
⎢
1
⎢ M
⎢
⎢
vm
⎢ v1v2
⎢
⎢ v1v3
⎢ M
⎢
G = ⎢ vm-1vm
⎢ v1v2v3
⎢
⎢ M
⎢ vm-2vm-1vm
⎢
⎢ M
⎢
v
⎢
1v2
Lvr
⎢ M
⎢
⎣vm-r
1Lv
v
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
or
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣v
m-1
m-2
+ m-1 m
m m-1 m-r+
1
where v = v ,v , L,
v )
i
(
i,0 i,1 i,n-1
v
v
m
v
v
v
v
m
m
v
v
v
m-1
v
v
v
2
0
m
M
M
1
m-1
m-2
v
M
M
1
v
Lv
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

Example:
Consider the 2 nd -order RM code for m=4,
n = 2
4 =
16
⎛4⎞
⎛4⎞
k = 1 + ⎜ ⎟ + ⎜ ⎟ = 11
⎝ 1⎠
⎝ 2⎠
n - k = 16 - 11 = 5
d
= 2
2
min
=
4
This is a (16, 11) linear code.
The generator matrix is given by
G
R(
2, 4)
⎡ v0
⎢
v
⎢
4
⎢ v3
⎢
⎢
v2
⎢ v1
⎢
= ⎢v4v
⎢v4v
⎢
⎢v4v
⎢v3v
⎢
⎢v3v
⎢
⎣v2v
3
2
1
2
1
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

where
v 0
= ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 )
v 4
= ( 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 )
v 3
= ( 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 )
v 2
= ( 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 )
v 1
= ( 0 10 1 0 10 1 0 10 1 0 10 1 )
v 4
v 3
= ( 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 )
v 4
v 2
= ( 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 )
v 4
v 1
= ( 0 0 0 0 0 0 0 0 0 10 1 0 10 1 )
v 3
v 2
= ( 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 )
v 3
v 1
= ( 0 0 0 0 0 10 1 0 0 0 0 0 10 1 )
v 2
v 1
= ( 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 )
⎛ m⎞
⎛m⎞
⎛m⎞
⎛ m⎞
Since k = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + L+
⎜ ⎟
⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ r ⎠
gives the number of information bits of the r-th order RM code of
length
m
2 , the information vector is generally expressed as
d
0 m m-1 1 m,m-1 m,m-2
dm,m-1,
,m-r+
= ( d d d
L
Ld
d d L
1)
For example, r=2, m=4, k=11
We have d = d d d d d d d d d d )
(
0 4 3 2 1 43 42 41 32 31
d21

The codewords are given by
c
= d ⋅G
Example: r=1, m=3
c = ( c c1
Lc7
) = ( d0
d3
d2
d1
0
)
⎡1
⎢
0
⎢
⎢0
⎢
⎣0
1
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1⎤
1
⎥
⎥
1⎥
⎥
0⎦