what is cyclic code ?

42510011 0010 1010 1101 0001 0100 1011

Wireless Communication Lab. CGU

What is Cyclic Code ?

指導教授：黃文傑 博士學生：吳濟廷2004.03.10

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OUTLINE

• Introduction• Encoding

– Generator polynomial

• Decoding– Parity check polynomial– Syndrome testing– Error correction

• Conclusion

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0011 0010 1010 1101 0001 0100 1011

4251

0011 0010 1010 1101 0001 0100 1011

Code Taxonomy

Today

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0011 0010 1010 1101 0001 0100 1011

Important cyclic code

• BCH code– Powerful random error-correcting codes

• Goley(23,12) code– Triple-error-correcting code

• RS code– Nonbinary, widely used for error control in data

communication and data storage

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0011 0010 1010 1101 0001 0100 1011

Cyclic Code• Definition

– Linearity: sum of any two codewords in the code is also a codeword

– Cyclic: any cyclic shift of a codeword in the code is also a codeword

a b c d ………………… x y z

z a b c d …………… w x y

1 cyclic shift

0 1 2 1( , , ,..., )nc c c c

1 0 1 2( , , ,..., )n nc c c c

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Cyclic shift

• How do we make such a shift cyclic?

2 10 1 2 1( ) ... n

nc X c c X c X c X

1 10 1 1 1

1 1 11 0 1 1

1 1 11 0 1 1

11

( ) ( ... ... )

... ...

... ...

( 1) ... ( 1)

i i n i n i nn i n i n

n n i i i nn i n n i

i i i nn i n n i

n i nn i n

X c X X c c X c X c X c X

c X c X c X c X c X

c c X c X c X c X

c X c X X

( ) ( )ic X

( )q X

( )( ) ( )( 1) ( )i n iX c X q X X c X

Code polynomial form

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0011 0010 1010 1101 0001 0100 1011

Generator polynomial

• be a polynomial of degree (n-k) that is a factor of

• Encoded codeword

where has degree (k-1)

( )g x

1nX

1

1

( ) 1n k

i n ki

i

g x g X X

general form :

( ) ( ) ( )c X a X g X( )a X

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0011 0010 1010 1101 0001 0100 1011

Encoding procedure

1.Multiply the message polynomial by

2.Divide by , obtaining the

3.Add to , obtaining the code polynomial

( )b X

( )m X n kX

( )n kX m X ( )g X ( )b X

( )n kX m X

( )c X

( ) ( )( )

( ) ( )

n kX m X b Xa X

g X g X

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0011 0010 1010 1101 0001 0100 1011

Encoding example

• Using the to generate a (7,4) systematic cyclic code from the message (1011)

3( ) 1g X X X

2 3( ) 1m X X X 3 2 3 3 5 6( ) (1 )n kX m X X X X X X X

Dividing by g(X)

3 5 6 2 3 3(1 )(1 ) 1X X X X X X X X quotient q(x) generator g(x) reminder3 3 5 6( ) ( ) ( ) 1c X p X X m X X X X

1001011U

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0011 0010 1010 1101 0001 0100 1011

EncoderEncoding (1 0 1 1) to a (7,4) cyclic code with 3( ) 1g X X X

(1 0 1 1)

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0011 0010 1010 1101 0001 0100 1011

Parity-check polynomial

• Definition

from linear block code ~

to check if the basis still has orthogonality

so we could get :

1

1

( ) 1k

i ki

i

h X h X X

0TGH

( ) ( ) mod( 1) 0

( ) ( ) 1

n

n

g X h X X

g X h X X

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0011 0010 1010 1101 0001 0100 1011

Syndrome testing

• Purpose : to check if there’s error in received codeword

• Result : syndrome polynomial

• How to do the syndrome testing ?

0,( )

0,s X

no error detected

error detected

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0011 0010 1010 1101 0001 0100 1011

Calculation of the syndrome

• Received codeword

where

( )r X1

0 1 1( ) ... nnr X r r X r X

generator polynomial

( ) ( ) ( ) ( )r X m X g X s X

syndromequotient

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Syndrome calculatorTransmitted codeword 0 1 1 1 0 0 1

Received codeword 0 1 1 0 0 0 1error

So we could get the

by syndrome calculator

What’s next ??

( ) 1s X X

For a (7,4) cyclic code

( 1 1 0 )

3( ) 1g X X X

syndrome calculator

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Error pattern

the code poly.

the Rx poly. (with error)

or in another form

so

( ) ( ) ( )c X a X g X

( ) ( ) ( )r X c X e X

( ) ( ) ( ) ( )r X m X g X s X

( ) ( ) ( ) ( ) ( ) ( )

( ) [ ( ) ( )] ( ) ( )

a X g X e X m X g X s X

e X a X m X g X s X

Syndrome of received codeword is exactly the same as syndrome of error pattern !!

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0011 0010 1010 1101 0001 0100 1011

Using the look-up table

Decoding table for (7,4) cyclic code

So from previous example,

we could get the location of error (0 0 0 1 0 0 0)

and get the corrected codeword

by adding the error pattern to received codeword which is

0 1 1 0 0 0 1

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

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Conclusion

• Cyclic code is good for error detection due to its structure

• It’s so easy to implement

• Convolution code next time ~

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0011 0010 1010 1101 0001 0100 1011

[Reference]

• Simon Haykin, ”Communication systems, 3rd ed.”, pp.686~pp699, John Wiley & Sons 1994

• Bernard Sklar, “Digital communications, 2nd ed.”, pp.356~pp.366, Prentice-Hall, 2002

• 盧而輝 ,”錯誤控制編碼” , pp.4.1~pp.4.34, 2001