13.CYclic Codes

7/28/2019 13.CYclic Codes

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Lecture 13

Systematic Cyclic Codes

Parity check polynomial


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Systematic Generator Matrix

Systematic Encoder copies message digits toconsecutive positions in codeword.

In each code word,

Rightmost k digits: Information digits

Leftmost n k digits: Parity check digits


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Generation of Systematic code

Let the message to be encoded is u= ( u0, u1, ..uk-1).The corresponding message polynomial is1

0 1 1

- 1 10 1 1

-

-

Multiplying ( ) by , we obtain a polynomial

of degree -1 or

( ) ...........

( ) .....

less

Dividing ( ) by the generator polynomialg( )

..

kk

n k n k n k nk

n k

n k

u x u u x u x

x u x u x u x

u x x

n

x x

x

u x

u

+

= + + +

= + + +


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Cond

10 1 1

Since the degree of g(x) is n

whe

-k,

( )a

the

nd ( )

degree

are quotient

of b(x) is

and remainder respectiv

n-k-1 or les

( ) ( ) ( ) ( )

ie.

ely

Rea

s

rrang

( ) ......

re

.

n k

n kn k

x u x a x g x b x

b x b

Wehav

b

e

a x

b

x

x

b

x

= +

= + + +

ing eqn.(1), we get the following polynomial

of degree n-1 or

( ) ( ) ( ) ( )

lessn kb x x u x a x g x+ =

This polynomial is a multiple of the generator polynomial g(x) andhence it is a code polynomial of the cyclic code generated by g(x).


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Cond

10 1 1

1 1

0 1 1

0 1 1 0 1

( ) ( ) ........

+ ........

( , ,....

Substituting the expressions,

This corresponds to the code vector,

.., , , ,

n k n kn k

n k n k n

k

n k

b x x u x b bx b x

u x u x u x

b b b u u

+

+ = + + +

+ + +

1....... )ku

Thus we generated a systematic code


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The n k parity check digits are the coefficients of the

remainder resulting from dividing the message

polynomial x n-ku(x) by the generator polynomial g(x)

To summarize, systematic encoding consists of the

following steps

Premultiply the message

Obtain the remainder [ ]

from dividing by the generator pol

( )

( )

( ) g( )

(

1:

2:

ynomial

Combine and to o3: btain the) ( )

n k

n k

n k

by

the parit

u x x

b x

x u x x

b x x u x

Step

Step y check digits

Step

code polynomial ( ) ( )n kb x x u x+


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Generator polynomial


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Parity-check polynomial


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Parity-check polynomial: check equations


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Syndrome polynomial


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Example of binary cyclic code


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Some properties Since g(x).h(x)=0 mod xn+1, g(x) and h(x) are

orthogonal

The polynomials xi g(x) and xj h(x) are also orthogonalfor all i and j

However, the vectors corresponding to g(x) and h(x)

are orthogonal only if the ordered elements of one ofthese vectors are reversed. This applies to xi g(x) andxj h(x) as well.

The (n, n-k) code generated by h(x) is just the same

as the code generated by the reciprocal polynomial ofh(x), except that the code words are reversed.


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13.CYclic Codes

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