Cyclic Codes

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By :Er. Amit Mahajan

CYCLIC CODES

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CYCLIC CODES



_Definition:



 An ( An (nn,, kk) linear code) linear code C C is called ais called a cyclic codecyclic code if every cyclic shift of a codeif every cyclic shift of a code

vector in

vector in C C is also a code vectoris also a code vector



 Codewords can be represented as polynomials of degreeCodewords can be represented as polynomials of degree nn. For a cyclic. For a cyclic

code all codewords are multiple of some

code all codewords are multiple of some polynomialpolynomial gg(( X X ) modulo) modulo X X nn₊₊₁₁

such that

such that gg(( X X ) divides) divides X X nn₊_+1._{1. g}_g(( X_{X ) is called the generator polynomial.}_{) is called the generator polynomial.}



Examples:



 Hamming codes, Golay Codes, BCH codes, RS codesHamming codes, Golay Codes, BCH codes, RS codes



 BCH codes were independently discovered by Hocquenghem (1959) andBCH codes were independently discovered by Hocquenghem (1959) and

by Bose and Chaudhuri (1960) by Bose and Chaudhuri (1960)



 Reed-Solomon codes (non-binary BCH codes) wReed-Solomon codes (non-binary BCH codes) were independently ere independently

introduced by Reed-Solomon introduced by Reed-Solomon

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Cyclic codes

are of interest and importance because

•

_{They posses rich algebraic structure that can be utilized in a}

variety of ways.

•

_{They have extremely concise specifications.}

•

_{They can be efficiently implemented using simple}

_{They can be efficiently implemented using simple shift}

_shift

registers.

•

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Notations

k = Number of binary digits i

k = Number of binary digits in the message before

n the message before

encoding

n = Number of binary digits in the encoded

message

n –

n – k =

k = r =

r = number

number of check

of check bits

bits

r r k k n n

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Basic properties of Cyclic codes:



 _{Let C be a binary (n,k) linear cyclic code}_{Let C be a binary (n,k) linear cyclic code}

1.

1. Within the set of code polynomials in C, there is a unique monicWithin the set of code polynomials in C, there is a unique monic

polynomial

polynomial with with minimal minimal degree degree is is called called the the generator generator polynomials.

polynomials.

2.

2. Every code Every code polynomial polynomial in in C, C, can can be be expressed expressed uniquely uniquely asas

3.

3. The generator The generator polynomial polynomial is is a a factor factor of of

4.

4. The orthogonality of The orthogonality of GG andand HH in polynomial form is expressed asin polynomial form is expressed as

.

. This

This means

means

is

is also

also a

a factor

factor of

of

)) (( X X g g r r _<_<nn.. _g_g((X X )) r r r r X X g g X X g g g g X X )) == ++ ++...++ (( 00 11 g g )) (( X X U U )) (( )) (( )) (( X X _m_m X X _g_g X X U U == )) (( X X g g 1 1 + + n n X X 1 1 )) (( )) (( X X hh X X == X X nn ++ g g hh(( X X )) X X nn ++11

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Polynomial Representation of Binary Information



_{It is convenient to think}

_{It is convenient to think of binary digits as}

_{of binary digits as}

coefficients of a polynomial in the

coefficients of a polynomial in the dummy

dummy

variable X.



_{Polynomial is}

_{Polynomial is written low-order-to-high-}

_written

low-order-to-high-order.

order.



_{Polynomials are treated according to the laws}

of ordinary algebra with an

of ordinary algebra with an exception addition

exception addition

is to be done

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Types of Cyclic Codes

There are two types of Cyclic Codes on basis of

Encoding:



_{Non-Systematic Encoding:}

Information bits are not packed together in the

codeword. These are rarely used.

c(x)= i(x) g(x)



_{Systematic Encoding:}

Information bits are packed together in the codeword.

c(x)= i(x) x

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8

Systematic Encoding



_{Systematic encoding algorithm for an (n,k)}

Cyclic code:

1.

Multiply the

Multiply

the message

message polynomial

polynomial

by

2.

Divide the result of Step 1 by the generator

polynomial

.

. Let

Let

be

be the

the remainder.

remainder.

3.

Add

to

to form

form the

the codeword

codeword

))

((

X X ii X X nn−−k k

))

((

X X

g

r r

((

X X

))

((

x x ii X X nn−−k k

))

((

X X C C

))

((

X X r r

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Implementation



_{The hardware to implement this algorithm is a shift}

register and a collection of modulo

register and a collection of modulo two adders.

two adders.



_{The number of shift register positions is equal to the}

degree of the divisor, G(X), and the dividend is shifted

through high order first and left to right.

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Circuit for encoding systematic cyclic codes

x

x _W_{We noticed earlier that}_{e noticed earlier that cyclic codes can}_{cyclic codes can be generated by using}_{be generated by using shift registers}_{shift registers}

whose feedback coefficients are determined directly by the generating whose feedback coefficients are determined directly by the generating polynomial

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Example

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Decoding cyclic codes

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 _{Every valid, received code word}_{Every valid, received code word} _R_R_(p)_{(p) must be a multiple of}_{must be a multiple of}_G_G_(p),_{(p), otherwise}_otherwise

an error has occurred. (Assume that the probability of noise to convert code an error has occurred. (Assume that the probability of noise to convert code words to other code words is very small.)

words to other code words is very small.)

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 _{Therefore dividing the}_{Therefore dividing the} _R_R_(p)/_(p)/_G_G_(p)_{(p) and considering the remainder as a}_{and considering the remainder as a}

syndrome can reveal if an error

syndrome can reveal if an error has happened and sometimes also to reveal inhas happened and sometimes also to reveal in which bit (depending on code strength)

which bit (depending on code strength)



 _{Division is accomplished by a shift registers}_{Division is accomplished by a shift registers}

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 _{The error syndrome of q=}_{The error syndrome of q=n-k}_{n-k bits is therefore}_{bits is therefore}

[ [

]]

(( ))

p

==

m

mo

od

d

(( )) //

p

(( ))

p

S

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Decoding cyclic codes: error correction

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Decoding circuit for (7,4) code

syndrome computation

x

x _{To start with, the switch is at “0” position}_{To start with, the switch is at “0” position}

x

x _{Then shift register is stepped until all the received code bits have entered the}_{Then shift register is stepped until all the received code bits have entered the}

register register x

x _{This results is a 3-bit syndrome (}_{This results is a 3-bit syndrome (n}_{n -- k}_{k = 3 ):}_{= 3 ):}

that is then left to the register that is then left to the register x

x _{Then the switch is turned to the position “1” that drives the syndrome out of}_{Then the switch is turned to the position “1” that drives the syndrome out of the}_the

register register

[ [

]]

(( ))

p

==

m

mo

od

d

(( )) //

p

(( ))

p

S

R

G

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Decoding

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Encoder and Decoder

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Advantages of Cyclic codes

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_{Simplified encoding.}



_{Easy syndrome calculation S(p) =}

_{Easy syndrome calculation S(p) = rem{Y(p)/G(p)}}

_{rem{Y(p)/G(p)}}

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_{Ingenious error correcting decoding methods have been}

_{Ingenious error correcting decoding methods have been devised for}

_{devised for}

specific cyclic codes. They eliminate the storage needed for t

specific cyclic codes. They eliminate the storage needed for table

able

lookup.

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_{Cyclic Redundancy Co}

_{Cyclic Redundancy Codes (CRC), a class of cyclic}

_{des (CRC), a class of cyclic codes, has ability to}

_{codes, has ability to}

detect burst errors of length ‘q’, all

detect burst errors of length ‘q’, all single bit errors, any odd number of

single bit errors, any odd number of

errors if (p+1) is a factor

errors if (p+1) is a factor of G(p), Double errors if G(p) has at

of G(p), Double errors if G(p) has at least

least

three 1’s.



_{Cyclic codes for error detection provides high efficiency and the ease of}

implementation.

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Some block codes that can be realized by cyclic codes

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 _(n,1)_(n,1) _{Repetition codes}_{Repetition codes}_{. High coding gain (minimum}_{. High coding gain (minimum distance always}_{distance always n}

n--1), but very low rate: 1/ 1), but very low rate: 1/nn

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 _(n,k)_(n,k) _{Hamming codes}_{Hamming codes}_{. Minimum distance always 3. Thus can detect 2}_{. Minimum distance always 3. Thus can detect 2}

errors and correct one error.

errors and correct one error. n=2n=2mm_{-1, k = n - m,}_{-1, k = n - m,}



 _{Maximum-length codes}_{Maximum-length codes}_{. For every integer there exists a}_{. For every integer there exists a maximum}_maximum

length code (

length code (n,k n,k ) with) with n = 2n = 2k k _{- 1,d}_{- 1,d} m

miinn = 2= 2kk--11 ..



 _BCH-codes_BCH-codes_{. For every integer there exist a}_{. For every integer there exist a code with}_{code with n}_{n = 2}_{= 2}mm_-1,_-1,

and where

and where t t is the error correction capabilityis the error correction capability



 _((n,k_n,k )) _{Reed-Solomon (RS) codes}_{Reed-Solomon (RS) codes}_{. Works with}_{. Works with k}_k_symbols_symbols _{that consists of}_{that consists of}

m

m bits that are encoded to yield code words of bits that are encoded to yield code words of nn symbolssymbols..

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 Nowadays BCH and RS are very popular due to large Nowadays BCH and RS are very popular due to large d d _m_miin_n , large number , large number of codes, and easy generation

of codes, and easy generation

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 _{Code selection criteria: number of codes, correlation properties, code}_{Code selection criteria: number of codes, correlation properties, code}