Linear block coding

LINEAR BLOCK CODING
Presented by
NANDINI MILI
JEEVANI KONDA

CONTENTS
Introduction
Linear block codes
Generator matrix
Systematic encoding
Parity check matrix
Syndrome and error detection
Minimum distance of block codes
Applications
Advantages and Disadvantages

INTRODUCTION
The purpose of error control coding is to enable the
receiver to detect or even correct the errors by introducing
some redundancies in to the data to be transmitted.
There are basically two mechanisms for adding
redundancy:
1. Block coding
2. Convolutional coding

LINEAR BLOCK CODES
The encoder generates a block of n coded bits from k information
bits and we call this as (n, k) block codes.
The coded bits are also called as code word symbols.
Why linear???
A code is linear if the modulo-2 sum of two code words is
also a code word.

 n code word symbols can take 2푛 possible values. From
that we select 2푘 code words to form the code.
 A block code is said to be useful when there is one to one
mapping between message m and its code word c as
shown above.

GENERATOR MATRIX
 All code words can be obtained as linear combination of basis vectors.
 The basis vectors can be designated as {푔1, 푔2, 푔3,….., 푔푘}
 For a linear code, there exists a k by n generator matrix such that
푐1∗푛 = 푚1∗푘 . 퐺푘∗푛
where c={푐1, 푐2, ….., 푐푛} and m={푚1, 푚2, ……., 푚푘}

BLOCK CODES IN SYSTEMATIC FORM
In this form, the code word consists of (n-k) parity check bits
followed by k bits of the message.
The structure of the code word in systematic form is:
The rate or efficiency for this code R= k/n

G = [ 퐼푘 P]
C = m.G = [m mP]
Message
part Parity part
Example:
Let us consider (7, 4) linear code where k=4 and n=7
m=(1110) and G = =
c= m.G = 풎ퟏ품ퟏ + 풎ퟐ품ퟐ + 풎ퟑ품ퟑ + 풎ퟒ품ퟒ
= 1.품ퟏ + ퟏ. 품ퟐ + ퟏ. 품ퟑ + ퟎ. 품ퟒ
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
품ퟎ
품ퟏ
품ퟐ
품ퟑ

c = (1101000) + (0110100) + (1110010)
= (0101110)
Another method:
Let m=(푚1, 푚2, 푚3, 푚4) and c= (푐1, 푐2, 푐3, 푐4, 푐5, 푐6, 푐7)
c=m.G= (푚1, 푚2, 푚3, 푚4)
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
By matrix multiplication we obtain :
푐1=푚1 + 푚3 + 푚4, 푐2=푚1 + 푚2 + 푚3, 푐3= 푚2 + 푚3 + 푚4, 푐4=푚1,
푐5=푚2, 푐6=푚3, 푐7=푚4
The code word corresponding to the message(1110) is (0101110) .

PARITY CHECK MATRIX (H)
 When G is systematic, it is easy to determine the
parity check matrix H as:
H = [퐼푛−푘 푃푇 ]
 The parity check matrix H of a generator matrix is
an (n-k)-by-n matrix satisfying:
T
퐻(푛−푘)∗푛퐺푛∗푘 = 0
 Then the code words should satisfy (n-k) parity
check equations
T T
푐1∗푛퐻푛∗(푛−푘) = 푚1∗푘퐺푘∗푛퐻푛∗(푛−푘) = 0

Example:
Consider generator matrix of (7, 4) linear block code
H = [퐼푛−푘 푃푇 ] and G = [ 푃 퐼푘]
The corresponding parity check matrix is:
H=
1 0 0
0 1 0
0 0 1
1 0 1
1 1 1
0 1 1
1
0
1
G. 퐻푇 =
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
ퟏ ퟎ ퟎ
ퟎ ퟏ ퟎ
ퟎ ퟎ ퟏ
ퟏ ퟏ ퟎ
ퟎ ퟏ ퟏ
ퟏ ퟏ ퟏ
ퟏ ퟎ ퟏ
= 0

SYNDROME AND ERROR DETECTION
 For a code word c, transmitted over a noisy channel, let r be
the received vector at the output of the channel with error e
+ c r = c+e
e
1, if r ≠c
0, if r=c i e =
Syndrome of received vector r is given by:
T
s = r.H =(푠1, 푠2, 푠3, … … . . , 푠푛−푘)

Properties of syndrome:
 The syndrome depends only on the error pattern and
not on the transmitted word.
T T T T
s = (c+e).H = c.H + e.H = e.H
 All the error pattern differ by atleast a code word
have the same syndrome s.

Example:
Let C=(0101110) be the transmitted code and r=(0001110) be the
received vector.
s=r. 퐻푇=(푠1, 푠2, 푠3)
=(푟1, 푟2, 푟3, 푟4, 푟5, 푟6, 푟7)
ퟏ ퟎ ퟎ
ퟎ ퟏ ퟎ
ퟎ ퟎ ퟏ
ퟏ ퟏ ퟎ
ퟎ ퟏ ퟏ
ퟏ ퟏ ퟏ
ퟏ ퟎ ퟏ
The syndrome digits are:
푠1 = 푟1 + 푟4 + 푟6 + 푟7 = 0
푠2 = 푟2 + 푟4 + 푟5 + 푟6 = 1
푠3 = 푟3 + 푟5 + 푟6 + 푟7 = 0

The error vector, e=(푒1, 푒2, 푒3, 푒4, 푒5, 푒6, 푒7)=(0100000)
*
C= r + e
= (0001110)+(0100000)
= (0101110)
where C *
is the actual transmitted code word

MINIMUM DISTANCE OFA BLOCK CODE
Hamming weight w(c ) : It is defined as the number of non-zero
components of c.
For ex: The hamming weight of c=(11000110) is 4
Hamming distance d( c, x): It is defined as the number of places where
they differ .
The hamming distance between c=(11000110) and x=(00100100) is 4
 The hamming distance satisfies the triangle inequality
d(c, x)+d(x, y) ≥ d(c, y)
 The hamming distance between two n-tuple c and x is equal to the
hamming weight of the sum of c and x
d(c, x) = w( c+ x)
For ex: The hamming distance between c=(11000110) and
is 4 and the weight of c + x = (11100010) is 4.
x=(00100100)

 Minimum hamming distance d : It is defined as the smallest
min
 The Hamming distance between two code vectors in C is equal
to the Hamming weight of a third code vector in C.
d = min{w( c+x):c, x €C, c≠x}
= min{w(y):y €C, y≠ 0}
= w
≠
min
distance between any pair of code vectors in the code.
For a given block code C, d is defined as:
dm i n=min{ d(c, x): c, x€C, c x}
min
min

APPLICATIONS
 Communications:
Satellite and deep space communications.
Digital audio and video transmissions.
 Storage:
Computer memory (RAM).
Single error correcting and double error detecting code.

ADVANTAGES DISADVANTAGES
 It is the easiest and
simplest technique to
detect and correct errors.
 Error probability is
reduced.
 Transmission bandwidth
requirement is more.
 Extra bits reduces bit rate
of transmitter and also
reduces its power.

Linear block coding

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