§6 linear codes § 6.1 classification of error control system § 6.2 channel coding conception §...
TRANSCRIPT
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§6 Linear Codes
§ 6.1 Classification of error control system
§ 6.2 Channel coding conception
§ 6.3 The generator and parity-check matrices
§ 6.5 Hamming codes
§ 6.4 Syndrome decoding
§6.1.1 Error control system
§6.1.2 classification of error correcting codes
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§6.1.1 Error control system
FECHECIRQ
ARQ
Automatic Repeat Request
Forward Error Correction
Hybrid Error Correction
Information Repeat Request
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§6.1.1 Error control system
1. ARQ (Automatic Repeat Request)
Error detecting codeword
verdict
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§6.1.1 Error control system
1. ARQ (Automatic Repeat Request)
1
1
2
ACK1
2х
NA
K2
2
2
AC
K2
3
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§6.1.1 Error control system
1 2 3 4 5 1
1 2 3 4 5х
NA
K1
2 3 4 5
1 2 3
6 7
х 4 5 6 7
3 4 5
1. ARQ (Automatic Repeat Request)
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§6.1.1 Error control system
1 2 3 4 5 1
1 2 3 4х
NA
K1
6 7
х
NA
K4
4
5 1 6 7
8
4
9
8
10
9
11
10х
12
1. ARQ (Automatic Repeat Request)
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Error correcting codeword
§6.1.1 Error control system
2. FEC (Forward Error Correction)
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codeword
3. HEC (Hybrid Error Correction)
§6.1.1 Error control system
verdict
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messages
messages
4. IRQ (Information Repeat Request)
§6.1.1 Error control system
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1. ClassificationError
correct
Non-linear codes
Linear codes
Block codesConvolution-al codes
Cyclic codes
Non-Cyclic codes
Correct random errors
Correct burst errors
Correct hybrid errors
§6.1.2 classification of error correcting codes
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2 0 1
3 0
c c c
c c
2 0 1
3 0 1
1c c c
c c c
linear codes and nonlinear codes
1. Classification
§6.1.2 classification of error correcting codes
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( 00 ) ( 10 ) ( 01 ) ( 11 )messages
( 001 )( 010 )( 100 )( 111 )forbidden words
( 000 ) codewords( 110 ) ( 101)
( 011)
(3,2) block codes
2 0 1c c c 2 1 0( )C c c c
Example 6.1.1
block codes and convolutional codes
§6.1.2 classification of error correcting codes
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jm 1jm 2jm in
out
(2,1,3)convolutional codes
Example 6.1.2 block codes and convolutional codes
The encoder for (3,2,2)convolutional codes
§6.1.2 classification of error correcting codes
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systematic codes and nonsystematic codes
Information sequence codeword
00 00000 01 10110 10 11011 11 01101
Information sequence codeword
00 00000 01 01011 10 10101 11 11110
1. Classification
§6.1.2 classification of error correcting codes
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• Exercise:
输入输出
)1(LC
)2(LC
)3(LC
+
M2
M1
M0
C0
C1
C2
C3
C4
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• Exercise:
M3
M2
M1
M0
C6
C5
C4
C3
C2
C1
C0
M3
M2
M1
C6
C5
C4
C3
C2
C1
C0
M0
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2. Error detecting codes
parity check codes
horizontal parity check codes
Example:
m=1101001001110011110000111
C = 101101111000011100010110110111
C’ =( transmitting by column) 101101111100000110100010011011101110
phalanx check codes
§6.1.2 classification of error correcting codes
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2. Error detecting codes
parity check codes
horizontal parity check codes
phalanx check codes
group counting codes
Information bits Check bits
111011 101 110110 100 111100 100 100011 011 000111 011 101001 011
§6.1.2 classification of error correcting codes
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2. Error detecting codes constant rate codes
Arabic numerals Fomer codes (3:2)codes
1 11101 01011 2 11001 11001 3 10000 10110 4 01010 11010 5 00001 00111 6 10101 10101 7 11100 11100 8 01100 01110 9 00011 10011 0 01101 01101
10
10
1
0
1
0
§6.1.2 classification of error correcting codes
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Review
• KeyWords:
ARQ FEC HEC IRQ
Linear block codes Convolutional codes
systematic codes
Error detecting codes
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Homework
1. List all (3:4) constant weight codewords .
2. If phalanx check code has error pattern as follows, can itdetect these errors?
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§6 Linear Codes
§ 6.1 Classification of error control system
§ 6.2 Channel coding conception
§ 6.3 The generator and parity-check matrices
§ 6.5 Hamming codes
§ 6.4 Syndrome decoding
§ 6.2 Channel coding conception
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1. General method for channel coding
Channel encoder
M=(mk-1,…,m0)
Redundant bits
Rules
(n>k)
C=(cn-1,…,c0)
§ 6.2 Channel coding conception
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§ 6.2 Channel coding conception
2. Decoding methods
C0
C1
┋
Ci
┋
C2k
-1
C0
C1
Ci
┋
Cj
┋
C2k
-1
C2k
┋
Ci’
┋
C2n
-1
Permittedcodewords
Forbidden vectors
Permittedcodewords
Transmitter
Receiver
Transmission mode map
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Decoding rules table:
2. Decoding methods
Subset m0 m1 m2 … m2k-1 Permitted
codes C0 C1 C2 C2k-1
)1(0C )1(
1C )1(2C )1(
12 kC
)2(0C )2(
1C )2(2C )2(
12 kC
┇ ┇ ┇ ┇
Forbidden codes
)12(0
knC )12(
1 kn
C )12(2
knC )12(
12
kn
kC
§ 6.2 Channel coding conception
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Example 6.2.1
2. Decoding methods
( 4 , 2 ) decoding table
Informtion 00 10 01 11
Codes 0000 1001 0111 1110
Forbidden
vectors
1000
0100
0010
0001
1101
1011
1111
0011
0101
0110
1010
1100
§ 6.2 Channel coding conception
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( 6 , 3 ) decoding table
§ 6.2 Channel coding conception
2. Decoding methods
message
codeword
one bit
error
two bits error
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1) Code rate
3. Parameters of coding
( n , k ) codes:
kR
n
§ 6.2 Channel coding conception
Example 6.2.2
( 23 , 12 ) BCH code:
120.52
23R
DVB-S2( 16200 , 10800 ) LDPC code:
108000.67
16200R
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Definition: The number of nonzero components in a codeword c.
2) Hamming weight
1 (001)c
2 (011)c
1( ) 1w c
2( ) 2w c
§ 6.2 Channel coding conception
3. Parameters of coding
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3) Hamming distance
Definition: The number of bit positions which are different between two codewords c and c’.
1 (101)c
2 (110)c 1 2( , ) 2d c c
§ 6.2 Channel coding conception
3. Parameters of coding
( , ) 0.( , ') 0 if
:
'.( , ') ( ', ).( , ') ( , '') ( '', ').
d c cd c c c cd c c d c cd c c d c c d
Properties
c c
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4) Minimum hamming weight
Definition: The minimum value of nonzero hamming weight in a code C.
}0,:)(min{)(min cCccwCw
0000 1001 0111 1110(4,2) code:
wmin=2
§ 6.2 Channel coding conception
3. Parameters of coding
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5) Minimum hamming distance
Definition: The minimum value of hamming distance in a code C.
},,:),(min{)(min ccCccccdCd
0000 1001 0111 1110(4,2)code:
dmin(C) = 2
§ 6.2 Channel coding conception
3. Parameters of coding
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Theorem 6.1: A code C={c1, c2, …, cM} is capable of detecting all error patterns of weight e iff dmin(C) e + 1.
§ 6.2 Channel coding conception
3. Parameters of coding
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Theorem 6.2: A code C={c1, c2, …, cM} is capable of correcting all error patterns of weight t iff dmin(C) 2t + 1.
(p147 theorem 7.2 in textbook)
§ 6.2 Channel coding conception
3. Parameters of coding
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Review
• KeyWords:
The method for codingrate
Hamming weight,
Capability of code
(encoding and decoding)
Minimum hamming weight
Hamming distance,Minimum hamming distance
(for detecting errors or for correcting errors )
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Homework
1. P163: T7.22 ;
2. Let code C={ (000000), (001110), (010101), (011011), (100011), (101101), (110110), (111000)} .
(1) calculate its minimum distance. (2) if it is used as error-detecting code, how many errors
can it detect?(3) if it is used as error-correcting code, how many errors
can it correct?
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Homework
3. Let C be an (n,k) code with minimum distance dmin, and let t≤e be nonnegative integers. Show that if dmin≥e+t+1, the C is t-error-correcting, e-error-detecting. (that is to say the errors within t can be corrected and meanwhile the e (t≤e ) errors can be detected.)
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§6 Linear Codes
§ 6.1 Classification of error control system
§ 6.2 Channel coding conception
§ 6.3 The generator and parity-check matrices
§ 6.5 Hamming codes
§ 6.4 Syndrome encoding§6.3.1 Definition of linear block codes
§6.3.2 The generator matrix
§6.3.3 The parity-check matrix
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§6.3.1 Definition of linear block codes
1. Vector spaceLet K be a field, a vector space over K is a set V with a special element 0(the zero vector) and with a vector addition denoted ‘+’ with the expected properties.
0 + v = v + 0 for all v V (property of 0)
v + w = w + v for all v,w V (commutativity)
(u +v) + w = u + (v + w) for all u,v,w V (associativity)
a(v +w) = av + aw for all v,w V, a K (distributivity)(a + b)v = av + bv for all v V, ab K (distributivity)a(bv) = (ab)v for all v V, ab K (associativity)
1*v = v for all v V (property of 1)
V
V
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1. Vector space
A linear combination of vectors v1, …, vn in a vector space V withcoefficients a1,…, an in the scalars K is the vector
nn vava ...11
The vectors v1,…,vn are linearly dependent if there is a set of coefficientsa1,…,an not all zero such that the corresponding linear combination is the zero vector: 0...11 nn vava
Conversely, vectors v1,…,vn are linearly independent if
0...11 nn vava
implies that all the coefficients ai are 0.
§6.3.1 Definition of linear block codes
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1. Vector space
Proposition: Given a basis e1,…,en for a vector space V, there is exactly one expression for an arbitrary vector v V as a linear combination of e1, …, en.
Definition: The dimension of a vector space to be the number of elements in a basis for it.
§6.3.1 Definition of linear block codes
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2. Linear block codesInformation
sequence codeword
00 11100 01 01011 10 10110 11 01101
4 1
3 0
2 1 0
1 1
0 0
C m
C m
C m m
C m
C m
Information sequence codeword
00 11000 01 10101 10 01110 11 00011
Example 6.3.1
§6.3.1 Definition of linear block codes
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2. Linear block codesExample 6.3.1
4 1 0
3 1
2 0
1 1 0
0 1
C m m
C m
C m
C m m
C m
Information sequence
codeword
00 00000 01 10110 10 11011 11 01101
4 1
3 0
2 1
1 0
0 1 0
C m
C m
C m
C m
C m m
Information sequence
codeword
00 00000 01 01011 10 10101 11 11110
C:
D:
§6.3.1 Definition of linear block codes
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2. Linear block codes
Definition: An (n, k) linear code over Fq is a k-dimensional
subspace of the n-dimensional vector space Vn,k(Fq)={(x1,…,xn): xi Fq}; n is called the length of the code, k the dimension. The code’s rate is the ratio k/n.
Encoder),,...,,( 0121 mmmm kk
),,...,,( 0121 cccc nn
(p140 definition in textbook)
§6.3.1 Definition of linear block codes
min
0
for a linear block code ,theorem 6 . there exists:
( ) min ( )
3
i
i
ix Cx
C
d C W x
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Example 6.3.2
2. Linear block codes
450
561
4562
463
ccc
ccc
cccc
ccc
Information sequence codewords
000001010
100011
101110111
00000000011101010011101110101001110101001111010011110100
(7,3) linear code
§6.3.1 Definition of linear block codes
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§6.3.2 The generator matrix
1. The generator matrix
1
2
1
0
k
k
g
g
G
g
g
1,0 1,1 1, 1
2,0 2,1 2, 1
1,0 1,1 1, 1
0,0 0,1 0, 1
k k k n
k k k n
n
n k n
g g g
g g g
g g g
g g g
1 1 2 2 1 1 0 0k k k kx m g m g m g m g
1
2
1 0 1 2 1 0
1
0
( ) ( )
k
k
n k k
g
g
c c m m m m
g
g
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§6.3.2 The generator matrix
1. The generator matrix
Definition: Let C be an (n,k) linear code over Fq . A matrix G whose row-space equals C is called a generator matrix for C. Conversely, if G is a matrix with entries from Fq, its row-space is called the code generated by G.
Example 6.3.3
A (5,1) linear code C1 with generator matrix.
G1 = [1 1 1 1 1].
(p140 definition in textbook)
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§6.3.2 The generator matrix
1. The generator matrix
Example 6.3.4
A (5,3) linear code C2 with generator matrix.
11111
01100
00111
2G
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§6.3.2 The generator matrix
1. The generator matrix
450
561
4562
463
ccc
ccc
cccc
ccc
0
1
2
0
1
2
3
4
5
6
110
011
111
101
100
010
001
m
m
m
c
cc
c
c
c
c
1011100
1110010
0111001
][ 0120123456 mmmccccccc
Gmmm 012
Example 6.3.2 (continued)
120
121
0122
023
04
15
26
mmc
mmc
mmmc
mmc
mc
mc
mc
120
121
0122
023
04
15
26
mmc
mmc
mmmc
mmc
mc
mc
mc
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§6.3.2 The generator matrix
1. The generator matrix
11 1
21 20
1
10 0
01 0
00 1
r
rk k r
k kr
p p
p pG I P
p p
For systematic codes:
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2. The encoder circuit of linear block codes
§6.3.2 The generator matrix
c6
c5
c4
c3
c2
c1
c0
1011100
1110010
0111001
G m2
m1
m0
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§6.3.3 The parity-check matrix
1. The parity-check matrix
01000110
00100011
00010111
00001101
0123456
0123456
0123456
0123456
ccccccc
ccccccc
ccccccc
ccccccc
450
561
4562
463
ccc
ccc
cccc
ccc
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§6.3.3 The parity-check matrix
1. The parity-check matrix
01000110
00100011
00010111
00001101
0123456
0123456
0123456
0123456
ccccccc
ccccccc
ccccccc
ccccccc
0
0
0
0
1000110
0100011
0010111
0001101
0
1
2
3
4
5
6
c
c
c
c
c
c
c
0 ( 0)T T THx or xH
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§6.3.3 The parity-check matrix
1. The parity-check matrix
Definition: Let C be an (n,k) linear code over Fq . A matrix H with the property that HxT = 0 iff x C is called a parity-check matrix for C.
(p142 definition in textbook)
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2. Relationship of G and H
§6.3.3 The parity-check matrix
1011100
1110010
0111001
G
1000110
0100011
0010111
0001101
H
(7,3) linear code
Example 6.3.2 (continued)
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2. Relationship of G and H
§6.3.3 The parity-check matrix
For systematic linear codes:
G = [Ik| P]
H = [PT| In-k]
0
0
T
T T
GH
or HG
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3. Shortened codes and dual codes
§6.3.3 The parity-check matrix
1) Shortened codes
C0=( 0000000) C4=( 1001011)C1=( 0010111) C5=( 1011100)C2=( 0101110) C6=( 1100101)C3=( 0111001) C7=( 1110010)
( 7, 3) linear block code:
(6,2)shortened code : C0 =( 000000 )C1 =( 010111 )C2 =( 101110 )C3 =( 111001 )
Example 6.3.5
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3. Shortened codes and dual codes
§6.3.3 The parity-check matrix
1) Shortened codes
1110100
0111010
1101001
)3,7(G )2,6(G
1000101
0100111
0010110
0001011
)3,7(H )2,6(H
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3. Shortened codes and dual codes
§6.3.3 The parity-check matrix
2) Dual codes
Example 6.3.5 (continued )
1000101
0100111
0010110
0001011
)3,7(H (7,4)
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Overview
• KeyWords:
Linear block code
The generator matrix
The parity-check matrix
Shortened codes and dual codes
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Homework
1. p159: T7.2 (a) ,(b)
2. A (7,4)linear code has generator matrix as follows:
0111000
1100100
1010010
1110001
G
(a)Plot the encoder circuit.(b)Find its parity-check matrix H.
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§6 Linear Codes
§ 6.1 Classification of error control system
§ 6.2 Channel coding conception
§ 6.3 The generator and parity-check matrices
§ 6.5 Hamming codes
§ 6.4 Syndrome decoding
§6.4.1 Standard array decoding
§6.4.2 Syndrome decoding
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§6.4.1 Standard array decoding
1. The standard array
Set of codewords
x ( 0 ) x ( 1 ) x ( 2 ) x ( 2k-1 )
(00…0)…
Subset …m0 m1 m2 m2k-1
Forbidden
vectors
E1 E1+x ( 1 ) E1+x ( 2 ) E1+x ( 2k-1 ) …
E2 E2+x ( 1 ) E2+x ( 2 ) E2+x ( 2k-1 ) …
E2n-k-1 E2n-k-1 +x ( 1 )E2n-k-1 +x ( 2 ) … E2n-k-1 +x ( 2k-1 )
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§6.4.1 Standard array decoding
1. The standard array 3 1
2 0
1 1 0
0 0
c m
c m
c m m
c m
Example 6.4.1:(4,2)linear code
1010
0111G
codewords x(0)
0000 x(1)
0111 x(2)
1010 x(3)
1101
Forbiddencodewords
1000 1111 0010 01010100 0011 1110 10010001 0110 1011 1100
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§6.4.1 Standard array decoding
2. The characters of standard array
The sum vector of two vectors in the same row is a codeword in the set of C.There are not the same n-dimensional vectors in the same
row. Each n-dimensional vector exists only in one row.
Theorem 6.4 Each (n,k) linear code is capable of correcting all of 2n-k error patterns.
(Prove it by yourself)
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000000 001110 010101 100011 011011 101101 110110 111000 S=000
000001 001111 010100 100010 011010 101100 110111 111001 001
000010 001100 010111 100001 011001 101111 110100 111010 010
000100 001010 010001 100111 011111 101001 110010 111100 100
001000 000110 011101 101011 010011 100101 111110 110000 111
010000 011110 000101 110011 001011 111101 100110 101000 011
100000 101110 110101 000011 111011 001101 010110 011000 110
001001 000111 011100 101010 010010 100100 111111 110001 101
§6.4.1 Standard array decoding
Example 6.4.2
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§6.4.2 Syndrome decoding
1. Syndrome vector
R=x+E
xHT=0
S=RHT=(sn-k-1…s1s0) =EHT
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§6.4.2 Syndrome decoding
1. Syndrome vector
Codewords C(0)
0000 C(1)
0111 C(2)
1010 C(3)
1101
S
00
1000 1111 0010 0101 10
0100 0011 1110 1001 11 Forbidden codewords
0001 0110 1011 1100 01
C(0)
0000 C(1)
0111 C(2)
1010 C(3)
1101
S
00
1000 1111 0010 0101 10
0100 0011 1110 1001 11
0001 0110 1011 1100 01
Theorem 6.5 2k codewords in each coset have the same syndrome, and different coset with different syndrome.
Example 6.4.1 ( continued ) (Prove it by yourself)
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000000 001110 010101 100011 011011 101101 110110 111000 S=000
000001 001111 010100 100010 011010 101100 110111 111001 001
000010 001100 010111 100001 011001 101111 110100 111010 010
000100 001010 010001 100111 011111 101001 110010 111100 100
001000 000110 011101 101011 010011 100101 111110 110000 111
010000 011110 000101 110011 001011 111101 100110 101000 011
100000 101110 110101 000011 111011 001101 010110 011000 110
001001 000111 011100 101010 010010 100100 111111 110001 101
§6.4.1 Standard array decoding
Example 6.4.2 ( continued )
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§6.4.2 Syndrome decoding
2. Syndrome decoding Example 6.4.3:
( 4 , 2 ) linear code syndrome decoding
Step1 : Computing the syndrome S=RHT
Step2 : getting E by S 令 E = (e3 e2 e1 e0) ,
3 2 1 0 1 0
1 0
1 1( ) ( )
1 0
0 1
TS EH e e e e s s
E Syndrome S
0000 00 1000 10 0100 11 0001 01
Step3 :x̂ R E
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S0
R=(r3r2r1r0)
c0 c1 c2 c3
r0 r1 r2 r3
S1
e0 e2 e3
Output in series
x=R+E^
§6.4.2 Syndrome decoding
2. Syndrome decoding
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r0 r1 rn-1
s0 s1 sr-1
e0 e1 en-1
c0 c1 cn-1
A general decoding circuit of (n,k) linear codes
S=RHt
S=EHt
^x=R+E
Received vectors storage
Compute syndrome circuit
Error pattern generator
n-step shift-register output
§6.4.2 Syndrome decoding
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Overview
• KeyWords:
Syndrome decoding
Standard array decoding
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Homework
1. p159: T7.2 (c) ,(e)
2. p159: T7.6
3. p159: T7.8
4. Give the standard array of (7,4) linear code with a parity matrix as follows:
1001011
0101101
0010111
H
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§6 Linear Codes
§ 6.1 Classification of error control system
§ 6.2 Channel coding conception
§ 6.3 The generator and parity-check matrices
§ 6.5 Hamming codes
§ 6.4 Syndrome decoding
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1. Relationship between H and S
§6.5 Hamming codes
Example 6.5.1: (7,3)linear code
0 6 5 4 3 2 1 0
1011000
1110100
1100010
0110001
H h h h h h h h
E = (e6 e5 e4 e3 e2 e1 e0)
S=RHT= EHT
3 6 4 3
2 6 5 4 2
1 6 5 1
0 5 4 0
s e e e
s e e e e
s e e e
s e e e
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S=RHT= EHT
1 1 1 2 10
1 2 0
1 2 0
n n
n n
rn rn r r n
h h h
H h h h
h h h
l et
1
2
j
j
j
rj
h
hh
h
1 0 1 2( ) (0 0 0)n i i itE e e e e e
1. Relationship between H and S
§6.5 Hamming codes
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1. Relationship between H and S1
21 0
0
[ ]
Tn
TT n
n
T
h
hS EH e e
h
1 1 2 2 0 0
1
T T Tn n n n
tT
ijj
e h e h e h
h
S is the linear
combination of columns of H
according to nonzero codebits
in E.
§6.5 Hamming codes
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1. Relationship between H and S
(p147 Theorem 7.3 in textbook)
§6.5 Hamming codes
Theorem 6.6 If all possible linear combinations of ≤ t columns of H are distinct, then dmin(C) ≥ 2t + 1, And so C can correct all error patterns of weight ≤t.
(p148 Corollary in textbook)Corollary If C is an (n,k) linear code with parity-check matrix H, dmin(C) = the smallest number of columns of H that are linearly dependent. Hence if every subset of 2t or fewer columns of H is linearly independent, the code is capable of correcting all error patterns of weight ≤ t.
Corollary : For a linear block code C(n, k) completely determined by its parity check matrix H, the minimum weight or minimum distance
of this code is equal to the minimum number of columns of that matrix which when added together result in the all-zero vector.
Corollary : Let a linear block code C have a parity check matrix H.
The minimum distance dmin of C is equal to the smallest positive number
of columns of H which are linearly dependent. That is, all combinations of
dmin - 1 columns are linearly independent, so there is some set of dmin columns
which are linearly dependent.
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2. Some bounds on error correcting codes
Theorem 6.7 The minimum distance for an (n, k) linear code is bounded by dmin (C)≤n-k+1.
Theorem 6.8 Let C is an (n,k) linear code, A t-random error correcting code C must have redundancy r(=n-k) satisfying:
1
2 1t
r in
i
C
§6.5 Hamming codes
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3. Hamming codes
Definition: Let H be an m ╳ (2m-1) binary matrix such that the columns of H are the 2m-1
nonzero vectors from Vm(F2) in some order. Then the n=2m-1, k=2m-1-m linear code over F2 whose parity-check matrix is H is called a (binary) Hamming code of length 2m-1.
(p149 Definition in textbook)
§6.5 Hamming codes
Length: n = 2m − 1Number of information bits: k = 2m − m − 1Number of parity check bits: n − k = mError-correction capability: t = 1, (dmin = 3)
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Example 6.5.2 : n=7
( 7 , 4 ) hamming codes
0001111
0110011
1010101
H
1 + 2→1
1 + 3→2
1 + 2 + 3→1
1101001
1011010
0111100
1H
1
1000011
0100101
0010110
0001111
G
1
0111100
1011010
1101001
H
2
1110100
0111010
1101001
H
2
1000101
0100111
0010110
0001011
G
§6.5 Hamming codes
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Encoding circuit
2
1000101
0100111
0010110
0001011
G
§6.5 Hamming codes
3. Hamming codes
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Decoding circuit
2
1110100
0111010
1101001
H
§6.5 Hamming codes
3. Hamming codes
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Overview
• KeyWords:
Hamming codes
The singleton bound
The Hamming bound
H, S, dmin, t
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Homework
1. p161: T7.17
2. p163: T7.23