CODE THEROY CYCLIC CODES

7/30/2019 CODE THEROY CYCLIC CODES

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Chapter 4

Linear Cyclic Codes

4.1 Definition of Cyclic Code

An ( n, k) linear code C is called a cyclic code if any cyclic shift of a

codeword is another codeword. That is,

if C c , , c , c c 1- n10 = )( L

then C c , , c , c , c c 2- n101- n1

= )()( L

Cyclic structure makes the encoding and syndrome computation

very easy.

In polynomial form

1- n1- n

2 210 x c x c x c c x c ++++= L)(

1- n 2- n

2101- n

1 x c x c x c c x c ++++= L)()(


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4.2 Generator Polynomial

Every nonzero code polynomial c( x) in C must have degree at

least n-k but not greater than n-1 .

There exists one and only one nonzero generator polynomial g( x)

for a cyclic code.

Uniqueness:

Suppose g( x) is not unique, then there would exist another such

code polynomial of the same degree of the form,

k- n1- k- n1- k- n

2 21 x x g' x g' x g'1 x g' +++++= L)( .

Thus one obtains the code polynomial

1- k- n1- k- n1- k- n11 x g g' x g g' x g" )()( ++= L)( .

Polynomial )( x g" corresponds to the word

))( 0 , g g' , , g g' , g g'0, 1- k- n1- k- n 2 211 (L .

Since k- n g" deg


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Every code polynomial c( x) is divisible by g( x), i.e. a multiple of

g( x).


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4.3 Selecting the Parameters of a Cyclic Code

It can be shown that the generator polynomial g( x) of an ( n, k)

cyclic code is always a polynomial factor of the polynomial 1- x n ,

(or 1 x n + ).

This means that any factor of 1- x n of degree n-k is a possible

generator of an ( n, k) cyclic code.

Since g( x) divides 1- x n , it follows that

)()( x g x h1- x n =

where k k 2

210 x h x h x h h x h ++++= L)(

where 1 h h k0 ==

i.e. k1- k1- k 2

21 x x h x h x h1 x h +++++= L)( .

h( x) is called the parity polynomial of an ( n, k) cyclic code.

Since )()( x g x h1- x n

= , one has the important relation

)()()( 1- x mod 0 x g x h n= .


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Define message polynomial as

1- k1- k

2 210 x m x m x m m x m ++++= L)(

Clearly, the product m( x) g( x) is the polynomial that represents

the codeword polynomial of degree n-1 or less.

An ( n, k) cyclic code is completely specified by the monic

generator polynomial

k- n1- k- n1- k- n1 x x g x g1 x g ++++= L)(

The codeword polynomial is expressed as

1- n1- n

2 210 x c x c x c c x c ++++= L)( .

1- n 2- n

2101- n

1 x c x c x c c x c ++++= L)()( .


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c( x) and )()( x c 1 are related by the formula

)()()()( 1- x mod x xc x c n1 =

This follows from the fact that since both c( x) and )()( x c 1 are

code polynomials

and

)()(

)(

)(

)(

1- x c x c

1- x c x c x c c

c- x c x c c

x c x c x xc

n1- n

1

n1- n

1- n 2- n01- n

1- n n

1- n01- n

n1- n0

+=

++++=

+++=

++=

LL

L

Now,

)()(

)()()()()(

)(

x g x m

1- x mod x g x m x x ci

nii

=

=

This identity proves that if any codeword is cyclically shifted i

times, that another codeword in the cyclic code C is formed.

Thus, any set of kq codewords which possesses the cyclic

property is generated by some generator polynomial g( x).


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4.4 Encoding of Cyclic Codes

Consider an ( n, k) cyclic code with generator polynomial g( x).

Suppose )( 1- k10 m , , m , m m L= is the message to be encoded.

1- k1- k10 x m x m m x m +++= L)( .

Multiplying m( x) by k- n x we obtain

1- n1- k

1 k- n1

k- n k- n x m x m x x m x +++= + L)(

Dividing )( x m x k- n by g( x), we have

)()()()( x p x g xq x m x k- n +=

where 1- k- n1- k- n10 x p x p p x p +++= L)( is the remainder.

Then, )()()()( x g xq x m x x p k- n =+ is a multiple of g( x) and

has degree n-1 . Hence it is the code polynomial for the message

m( x).


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Note that

4 4 4 4 4 4 34 4 4 4 4 4 21L

4 4 4 4 34 4 4 4 21L

messageparity

)()(1- n

1- k1 k- n

1 k- n

01- k- n

1- k- n10

k- n

x m x m x m x p x p p

x m x x p

+++++++=

+

+

The code polynomial is in systematic form where p( x) is the

parity-check part.

The encoding can be implemented by using a division circuit

which is a shift register with feedback connections based on the

generator polynomial g( x), as shown below.

Figure 4.2.

Note that the right-most symbol of the word is the first symbol to

enter the encoder, subscript-order notwithstanding.


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The gate is turned on until all the information digits have been

shifted into the circuit.

Note that

1- n1-i- n

1i1

i0

1-i1- n1i- ni- n

i

x c x c x c

x c x c c x c

++++

+++=

+

+

L

L)()(

and 1-i n1- n1i

1i

0i x c x c x c x c x ++ +++= L)(

Thus

)())((

)()()(

)(

)( x c1- x xq

1- x c1- x c1- x c

x c x c x c x c c x c x

i n

n1- n

n1i- n

ni- n

1- n1-i- n

i0

1-i1- n1i- ni- n

i

+=

++++

++++++=

+

+

L

LL

Syndrome circuit with input from the right end. ( Fig. 4.5(b) )

)( x r


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After the entire r ( x) has been shifted into the register, the

contents in the register do not form the syndrome of r ( x);

Rather, they form the syndrome )()( x s k- n of )()( x r k- n , which is

the cyclic shift of r ( x).

This is because shifting r ( x) from the right end is equivalent to

premultiplying r ( x) by k- n x .

Since)())(()(

)()()()()( x r1- x x b x r x

x x g x a x r x k- n n k- n

k- n

+=

+=

and )()( x g x h1- x n =

we have )()()]()()([)()( x x g x a x h x b x r k- n ++=

This says that, when )()( x r k- n is divided by g( x), )( x is also

the remainder. Therefore )( x is )()( x s k- n .


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Example 4.9 (page 153)

For a ( 7, 4) code, 3 x x1 x g ++=)(

)()( 0101100 c0101 m ==

Figure 4.3 Encoder for the ( 7, 4 ) cyclic code generated by3 x x1 x g ++=)( .

shift number i register contentsR 1 R 2 R 3

switch gate output

0 0 0 0 1 on 01 0 0 0 1 on 102 1 1 0 1 on 0103 0 1 1 1 on 10104 0 0 1 2 off 110105 0 0 0 2 off 0110106 0 0 0 2 off 0011010

Table 4.5 Encoding operation for the (7, 4) cyclic encoder shown in Fig. 4.3.


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Code generation by using parity polynomial h ( x)

k k

2 210 x h x h x h h x h ++++= L)(

with 1 h h k0 ==

We want to show that a parity-check matrix C may be obtained

from h ( x).

Let )( 1- n10 c , , c , c c L= be a code vector of C

Then )()()( x g x m x c =

Multiply c( x) by h ( x), we obtain

)()(

))((

)()()()()(

x m x m x

1- x x m

x h x g x m x h x c

n

n

=

=

=

Since the degree of m( x) is k-1 or less, the powers

1- n1 k k x , , x , x L+ do not appear in )()( x m x m x n . If we expand

the product c( x) h( x) on the LHS of the above equation, the

coefficients of 1- n1 k k x , , x , x L+ must be equal to zero. Therefore,

we obtain the following n-k equalities:

k- n j10 c h k

0i j-i- ni

==

for (4.22)


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Since 1 h k = , the above equation (4.22) can be put into the

following form:

k- n j1 c h c1- k

0i j-i- ni j- k- n

= =

for (4.24)

For a cyclic code in systematic form, the components

1- n1 k- n k- n c , , c , c L+ of each code vector are information (message)

digits.

Given these k message digits, the above equation is a rule to

determine the n-k parity-check digits, 1- k- n10 c , , c , c L .

An encoding circuit (with k-stage shift register) based on the

above equation is shown below. (Fig. 4.4)

Figure 4.4 A k-stage shift-register encoder for an ( n, k ) cyclic code generated by

the parity-check polynomialk

k

2

210 x h x h x h h x h++++= L

)( .


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4.5 Syndrome Computation

Let c( x) and r ( x) be the transmitted code polynomial & received

polynomials, respectively.

Dividing r ( x) by the generator polynomial g( x), we have

)()()()( x s x g xq x r +=

where k- n1- k- n1- k- n 2

210 x x g x g x g g x g +++++= L)(

and 1- k- n1- k- n 2

210 x s x s x s s x s ++++= L)( is the remainder.

Then s( x) is the syndrome of r ( x).

r( x) is a code polynomial if and only if s( x) = 0.

Syndrome computation is done by a division circuit as shown

below:

)( x r


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As soon as the entire r( x) has been shifted into the register, the

contents in the register form the syndrome s( x).

Since )()()( xe x c x r +=

and also )()()()( x s x g xq x r +=

we have

)()()]()([

)()()()()(

)()()()(

)()()(

x s x g x m xq

x g x m x s x g xq

x c x s x g xq

x c x r xe

++=

++=

++=

+=

or )()()( x g mod xe x s =

Hence the syndrome polynomial s( x) is also the remainder that

results from dividing e( x) by g( x).


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4.6 Error Detection and Syndrome Decoding

Cyclic codes are very effective for error detection.

Suppose r ( x) is the received polynomial,

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

)(-)(

)()(

x h x g r x s x g xq x

1- x r x xr

x r x r x r r x r

1- n

n1- n

1- n 2- n

2101- n

1

+=

=

++++= L

(4.33)

Also denote that )()()()( )()( x s x g x a x r 11 +=

Where )()( x s 1 is the remainder that results from dividing

)()( x r 1 by g( x), i.e. the syndrome polynomial of the

cyclically-shifted received word )()( x r 1 .

From (4.33)

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

)()(-)()()()()()()(

)(

x s x g x h r x xq x a

x g x xq x h x g r x s x g x a x xs1

1- n

1- n1

++=

++=

Thus, )()( x s 1 is also the remainder which results from the

division of xs( x) by g( x).

That is, if )()( x r 1 is the cyclic shift of r ( x) one step to the right,

the syndrome )()( x s 1 which corresponds to )()( x r 1 satisfies

)()()()( x g mod x xs x s 1 = ( 4 . 35 )


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Equation (4.35) generalizes easily to the i-th cyclical shift case.

That is, the remainder obtained by dividing )( x s x i by g( x) is

the syndrome )()( x s i corresponding to )()( x r i .

Example:

Consider the syndrome circuit for ( 7, 4) cyclic code generated by

3 x x1 x g ++=)( .

The code has 3 d min = and is capable of correcting any single

error in e( x).

Since the error patterns consist of the cyclic shifts of

(0 0 0 0 0 0 1 ), the decoder needs only to be able to recognize one

of the seven nonzero syndromes to be able to correct all of the

nonzero error patterns.

The syndrome s = (1 0 1), corresponding to the error pattern e =

(0 0 0 0 0 0 1), is the best choice since it allows one to release the

corrected codeword bits before the error location actually is

identified.


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Fig. 4.6. (page 160)

)( x r)( x c

Decoding begins by first setting all of the shift-register cells to

zero. The received word r is then shifted bit-by-bit into the

7 -bit received word buffer and the syndrome-computation circuit,

simultaneously.

Once the received word is completely shifted into the buffer, the

shift-registers of the syndrome computation circuit contain the

syndrome for the received word. As one continues to shift

cyclically the contents of the received-word buffer and the

syndrome computation circuit, the syndrome computation circuit

computes the syndrome for the cyclically shifted versions of the

received word.


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If at any point the computed syndrome is s = (1 0 1), i t is

detected by the AND gate when its output goes to 1. This value

is then used to complement and correct the error in the rightmost

bit in the buffer as it leaves the buffer.


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A general decoder (Meggitt decoder) for an ( n, k) cyclic code is

shown below. (Lin / Costello p. 105)

The decoding operation is described as follows:

1. Shift the received polynomial r( x) into a buffer and the

syndrome registers simultaneously. (That is, syndrome is

formed.)

2. Check whether the syndrome s( x) corresponds to a correctable

error pattern 1- n1- n10 xe xee xe +++= L)( with an error at

the highest-order position 1- n x . (i.e. 1e 1- n = ?)

3. Correct 1- n r if 1e 1- n = .

4. Cyclically shift the buffer and syndrome registers once

simultaneously. Now the buffer register contains )()( x r 1 and

the syndrome register contains the syndrome )()( x s 1 of

)()( x r 1 .

5. Check whether )()(

x s1

corresponds to a correctable error

pattern )()( xe 1 with an error at the highest-order position

1- n x .

6. Correct 2- n r if it is erroneous.

7. Repeat the same process until n shifts.


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8. If the error pattern is correctable, the buffer register contains

the transmitted codeword and the syndrome register contains

zeros.

9. If the syndrome register does not contain all zero at the end of

the decoding process, an uncorrectable error pattern has been

detected.

)( x r i r

ie

Figure 4.8 General cyclic code decoder with received polynomial r( x) shifted into

the syndrome register from left end.


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4.7 Shortened Cyclic Codes

Suppose we wish to have information blocks of ( k-l ) digits, but we

use ( n, k) code with l < k.

Let )(876

LL

l

l -1- k10 000 , m , , m , m m =

If the code is a cyclic code in systematic form, the code words

generated are )(876

LLL

l

1-l - k101- k- n10 000 , m , , m , m , c , , c , c c =

There is no need to transmit the l zeros, since the receiver will

know this (because we are designing it). The code length is

l - n n' = .

In general, a code obtained from shortening a cyclic code is no

longer cyclic.

Shortened codes have error detection and correction capabilities

that are at least as good as the original codes.


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Example:

Consider the ( 7, 4) cyclic code generated by 3 x x1 x g ++=)(

=

1000100

0100111

0010110

0001011

G

The generator matrix of the ( 6, 3) shortened code is obtained by

omitting the last row and column from G of the original ( 7, 4)

code.

=

100111

010110

001011

G'

and the parity-check matrix H' is obtained by omitting the last

column of H .

=

110100

111010

011001

H'

Decoding of shortened code

A shortened cyclic code ( n-l, k-l ) can be decoded by a decoder

similar to the original ( n, k) cyclic code. (Fig. 4.4)


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Let 1-l - nl -1- n10 x r x r r x r +++= L)( be received polynomial.

Suppose that r ( x) is shifted into the syndrome register from the

right end. (Fig. 4.5(b))

If the decoding circuit for the original cyclic code is used for

decoding the shortened code, the proper syndrome for decoding

the received digit 1-l - n r is equal to the remainder resulting from

dividing )( x r x l - k- n by the generator polynomial g( x).

Since shifting r( x) into the syndrome register from the right end

is equivalent to premultiplying r ( x) by k- n x , the syndrome

register must be cyclically shifted for another l times after the

entire r ( x) has been shifted into the register.

These extra l shifts can be eliminated by modifying the

connection of the syndrome register:

Dividing )( x r x l - k- n by g( x), we obtain

)()()()( x s x g x a x r xl - k- n

1l - k- n

+=

Next, we divide l - k- n x by g( x).

Let 1- k- n1- k- n10 x x x +++= L)( be the remainder resulting

from this division. That is )()()( x g x a x x 2l - k- n +=

and then )()()]()()([)()()(

x s x g x r x a x a x r xl - k- n

21++=


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The equation above suggests that we can obtain the syndrome

)()( x s l - k- n by multiplying r ( x) by )( x and dividing the product

)()( x r x by g( x).

Computing )( x s l - k- n this way, the extra l shifts of the syndrome

register can be avoided.

The circuit is shown below. (Lin / Costello, p. 117)

0 1 2 2- k- n 1- k- n

1- k- ng 2- k- ng 2g1g

Figure 4.15 Circuit for multiply by and dividing by

1- k- n1- k- n10 x x x +++= L)( and dividing )()( x r x by

k- n10 x x g g x g +++= L)(


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4.8 CRC Codes

Cyclic Redundancy Check (CRC) Codes are error-detecting codes

typically used in automatic repeat request (ARQ) systems.

CRC codes have no error-correction capability but they can be

used in combination with an error-correcting code to improve the

performance of the system.

CRC codes are shortened cyclic codes.

Concatenated Coding System


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CRC codes are often (but not always) constructed of polynomials

of the form )()()( x p1 x x g +=

where p( x) is a primitive polynomial of degree ( r-1 ) that divides

1- x1- r 2 .

16 -bit CRC-CCITT

1 x x x x g 51216 +++=)(

The 16 -bit CRC-CCITT is defined as the remainder obtained by

dividing the message polynomial 16 x x m )( by g( x).

The codewords are formed as a k-bit information sequence

followed by a check pattern of 16 bits (two 8-bit bytes) at the end

of message. The result is a ( k+16, k ) code.


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Fig. 4.9. (page 178)

Figure 4.9 Systematic encoder for CCITT polynomial code with

1 x x x x g 51216 +++=)( .

In the encoding process, two zero bytes are added to the end of

the message, which are used when computing the CRC.

At the receiver the decoder simply computes the CRC of the

message part and adds the results to the CRC bytes, and then

tests to see whether the result equals zero.


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A CRC constructed by an ( n, k) cyclic code is capable of detecting

any error burst of length n-k or less.

Also, the fraction of undetectable error burst of length n-k+1 is

)( 1 k- n- 2 + .

CRC in spread spectrum cellular system

1 x x x x x

x x x x x x x x x g 26 7 8

11121315 20 21 29 30

++++++

+++++++=)(

CRC used in ATM protocol (CRC-ATM)

1 x x x x g 28 +++=)(

CODE THEROY CYCLIC CODES

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