1. LINEAR BLOCK CODING
Presented by:
Manish Srivastava

2. LINEAR BLOCK CODE
In a (n,k) linear block code:
1st portion of k bits is always identical to the
message sequence to be transmitted.
2nd portion of (n-k ) bits are computed from message
bits according to the encoding rule and is called
parity bits.

3. SYNDROME DECODING
 The generator matrix G is used in the encoding
operation at the transmitter
 The parity- check matrix H is used in the
decoding operation at the receiver
 Let , y denote 1-by-n received vector that
results from sending the code x over a noisy
channel
y=x +e

4.  For i=1,2,….., n
ei= 1,if an error has occurred in the ith location
0 ,otherwise
o s=yHt

5. PROPERTIES
Property 1:
 The syndrome depends only on the error
pattern and not on the transmitted code
word.
 S=(x+e)Ht
=xHt+ eHt
=eHt

6. PROPERTY 2:
 All error pattern that differs at most by a code
word have the same syndrome.
 For k message bits ,there are 2k distinct codes
denoted as xi ,i=0,1, ………. 2k -1
we define 2k distinct vectors as
e =e+ xi i=0,1,…….. 2k-1

7. =e +
=e

8. PROPERTY 3:
 The syndrome s is the sum of those columns of
matrix H corresponding to the error locations
H=[ , ………., ]
therefore,
s=

9. PROPERTY 4:
 With syndrome decoding ,an (n,k) linear block
code can correct up to t errors per code word
,provided that n and k satisfy the hamming
bound
≥ ( )
 where ( ) is a binomial coefficient ,namely
( )= n!/(n-i)!i!

10. MINIMUM DISTANCE CONSIDERATIONS:
 Consider a pair of code vectors x and y that
have the same number of elements
 Hamming distance d(x,y): It is defined as the
number of locations in which their respective
elements differ .
 Hamming weight w(x) : It is defined as the
number of elements in the code vector.

11.  Minimum distance dmin: It is defined as the
smallest hamming distance between any pair of
code vectors in the code or smallest hamming
weight of the non zero code vectors in the code
.

12.  An (n,k) linear block code has the power to
correct all error patterns of weight t or less if
,and only if
d( ) ≤2t+1
 An (n,k) linear block code of minimum distance dmin
can correct upto 1 error if and only if
t≤ [1/2 (dmin – 1)].

13. Advantages Disadvantages
Easiest to detect and  Transmission
correct errors. bandwidth is more.
Extra parity bit does not  Extra bit reduces the
convey any information bit rate of transmitter
but detects and and also its power.
corrects errors.

14. APPLICATIONS
 Used for error control coding.
 Storage-magnetic and optical data storage in hard
disks and magnetic tapes and single error
correcting and double error correcting code(SEC-
DEC) used to improve semiconductor memories.
 Communication-satellite and deep space
communications.

15. THANK YOU!!