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II. Linear Block Codes



© Tallal Elshabrawy 2

Digital Communication Systems

Source ofInformation

User ofInformation

Source

Encoder

ChannelEncoder

Modulator

Source

Decoder

ChannelDecoder

De-Modulator

Channel




What are Linear Block Codes?

Information sequence is segmented into message blocks of

f ixed leng th .

Each k-bit information message is encoded into an n-bit

codeword (n>k )

Linear Block Codes

Binary BlockEncoder2

k

k -bit Messages 2k

n -bit DISTINCTcodewords




Linear Independence

A set of vectors g0, g1,…, gk -1 are linearly independent if

there exists no scalars u0, u1,…, uk-1 that satisfy

0 1 10 1 k-1v g g g 0k u u ... u

Unless u 0=u 1=…= u k- 1=0

Examples

[0 1 0 ], [1 0 1], [1 1 1] are ……… Linearly Dependent

[0 1 0 ], [1 0 1], [0 0 1] are ………

Linearly Independent




Why Linear?

Encoding Process

Store and Index 2k

codewords of length n Complexity

Huge storage requirements for large k

Extensive search processing for large k

Linear Block Codes Stores k linearly independent codewords

Encoding process through linear combination of codewords g0,g1,…, gk - 1 based on input message u=[u0 , u1,…, uk-1]

00 01 0 10

10 11 1 11

1 0 1 1 1 11

g

g

.G=

.

.

g

,n

,n

k , k , k ,n k

g g ... g

g g ... g

. . .

. . .

. . .

g g ... g

Generator Matrix

v=u.G


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Example

0

1

3

1 1 0 1 0 0 0

0 1 1 0 1 0 0

1 1 1 0 0 1 0

1 0 1 0 0 0 1

2

g

gG=

g

g

Message Codeword

0000 0000000

0001 1010001

0010 1110010

0011 0100011

0100 0110100

0101 1100101

0110 10001100111 0010111

1000 1101000

1001 0111001

1010 0011010

1011 1001011

1100 1011100

1101 0001101

1110 0101110

1111 1111111

g0

g1

g2

g3

u= [0 1 1 0]

Linear BlockEncoder (v=u.G)

v= g1+g2 v= [1 0 0 0 1 1 0]



Example

0

1

3

1 1 0 1 0 0 0

0 0 1 0 1 1 1

1 1 1 1 1 1 1

1 0 1 0 0 0 1

2

g

gG=

g

g

u= [0 1 1 1]

Block Encoder(v=u.G)

v= g1+g2+g3

v= [0 1 1 1 0 0 1]

LinearlyDependent

u= [1 0 0 1]

Block Encoder(v=u.G)

v= g0+g3 v= [0 1 1 1 0 0 1]

NOT DISTINCT



Linear Systematic Block Codes

Redundant CheckingPart

MessagePart

n -k bits k bits

00 01 0 1

10 11 1 1

1 0 1 1 1 1

1 0 0

0 1 0

0 0 1

G= P I

,n k

,n k

k

k , k , k ,n k

p p ... p ...

p p ... p ...

. . . . . .

. . . . . .

. . . . . .

p p ... p ...

p-matrix kxk- identity matrix



The Parity Check Matrix

For any k x n matrix G with k linearly independent rows,

there exists an (n-k ) x n matrix H (Parity Check Matrix),such that

G.HT=0

00 01 1 0

01 11 1 1

0 1 1 1 1 1

1 0 0

0 1 0

0 0 1

T

H= I P

k ,

k ,

k

,n k ,n k k ,n k

p p ... p ...

p p ... p ...

. . .. . .

. . .. . .

. . .. . .

p p ... p ...



Example

1 1 0 1 0 0 00 1 1 0 1 0 0

1 1 1 0 0 1 0

1 0 1 0 0 0 1

G

1 0 0 1 0 1 10 1 0 1 1 1 0

0 0 1 0 1 1 1

H



Encoding Circuit 1 1 0 1 0 0 0

0 1 1 0 1 0 0

1 1 1 0 0 1 0

1 0 1 0 0 0 1

G

u 0 u 1 u 2 u 3

Input u

To channel

+ + +

v 0 v 1 v 2

Parity Register

Message Register

[u0 u1 u2 u3] [v0 v1 v2 u0 u1 u2 u3]

Output v

Encoder Circuit



Syndrome

Characteristic of parity check matrix (H)

Tv.H 0

Tv.H 0

v C

v C

Channelv r

+v r=v+e

e Error Pattern

Syndrome

Ts=r.H



Error Detection

s 0 r Cr is NOT a codeword

An Error is Detected: What Options do we have? Ask for Retransmission of Block

Automatic Repeat Request (ARQ)

Attempt the Correction of Block

Forward Error Correction



Undetectable Error Patterns

Can we be sure that r=v ?? NO! WHY?

s 0 r C

e C r C

How many undetectable error patterns exist? 2k -1 Nonzero codeword means

2k -1 undetectable error patterns



Syndrome Circuit

1 0 0

0 1 0

0 0 1

1 1 0

0 1 1

1 1 1

1 0 1

T

H

0 0 3 5 6

1 1 3 4 5

2 2 4 5 6

s r r r r

s r r r r

s r r r r

r 1 r 2 r 3 r 4 r 5 r 6 r 0

+ + +

s 0 s 1 s 2




Hamming Weight and Hamming Distance

Hamming Weight w(v):

The number of nonzero components of v

Hamming Distance d (v,w):

The number of places where v and w differExample:

v= (1 0 1 1 1 0 1): w(v)=5

w=(0 1 1 0 1 0 1): d (v,w)=3

Important Remark: In modulo(2)

d (v,w)=w(v+w)

In the example above v+w=(1 1 0 1 0 0 0)




Minimum Distance of a Block Code

Minimum Distance of a block code (dmin

):

The minimum Hamming distance between any two

code words in the code book of the block code

mind min v, w : v, w C, v w d

w

w

min

min

min min

d min v w : v, w C, v w

d min x : x C, x 0

d w (minimum weight of the code)



© Tallal Elshabrawy 2006-02-16Lecture 9 20

Linear block codes – cont’d

Standard array and syndrome tabledecoding

1. Calculate

2. Find the coset leader, , corresponding to .

3. Calculate and corresponding .

Note that

If , error is corrected.

If , undetectable decoding error occurs.

T rHS

iee ˆ S

erU ˆˆ m̂

)ˆˆ(ˆˆ e(eUee)UerU ee ˆ

ee ˆ





Example: Standard array for the (6,3) code

010110100101010001

010100100000

100100010000

111100001000

000110110111011010101101101010011100110000000100000101110001011111101011101100011000110110000010

000110110010011100101000101111011011110101000001

000111110011011101101001101110011010110100000000

Coset leaders

coset

codewords





111010001

100100000

010010000

001001000

110000100

011000010

101000001

000000000

(101110)(100000)(001110)ˆˆ

estimatedisvectorcorrectedThe

(100000)ˆ

issyndromethistoingcorrespond patternError

(100)(001110):computedisof syndromeThe

received.is (001110)

ted. transmit(101110)

erU

e

HrHSr

r

U

T T

Error pattern Syndrome




Hamming Codes

For any positive integer m≥3, there exists a Hamming

code such that:

- Code Length: n = 2m-1

- No. of information symbols: k = n-m =2

m

-m-1- No. of parity check symbols: n-k = m

- Error correcting capability: t = 1 (i.e., dmin=3)




Example: (15,11) Hamming Codes

- Code Length: n = 24

-1 = 15- No. of information symbols: k = 15-4 = 11

- No. of parity check symbols: 15-11 = 4

- Error correcting capability: dmin = 3

1 0 0 0 1 1 1 0 0 0 1 1 1 0 1

0 1 0 0 1 0 0 1 1 0 1 1 0 1 1H0 0 1 0 0 1 0 1 0 1 1 0 1 1 1

0 0 0 1 0 0 1 0 1 1 0 1 1 1 1

4 columns ofweight 1

11 columns ofweight >1



© T ll l El h b 26

- Number of cosets = 2n-k= 215-11 = 16

- Number of error patterns of weight 1 = 15

- If the coset leaders are chosen to be the 0 vector and all

the error patterns of weight 1. The (15,11) code corrects

ONLY error patterns with single error

The (15,11) Hamming Code is a Perfect Code

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