16.reed solomon codes

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Lecture 16

Non-Binary BCH Codes


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Introduction

For any choice of positive integers s

and t, there exists a q-aryBCH codeof length n=qs -1

This code is capable of correcting anycombination oft or fewer errors

This requires no more than 2st parity

check digits


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Generator Polynomial

Let be a primitive element in GF(qs).The generator polynomial g(x) of a t

error correcting q-aryBCH code is thepolynomial of lowest degree withcoefficients from GF(q) for which

are roots.

Let

Then

2 2, ,......., t ( ) be the minimal polynomial of .ii x

1 2 2( ) LCM{ ( ), ( ),......., ( )}tg x x x x =


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The degree of each minimal polynomialis s or less. Hence the degree ofg(x)

is atmost 2st.

Therefore the no. of parity check digits

of the code generated by g(x) is nomore than 2st.

q=2 corresponds to binary BCHcode.


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The special sub-class ofq-ary BCHcodes for which s=1 is the most

important type ofq-aryBCH code.

These codes are called Reed-Solomoncodes in honor of their discoverers.


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Reed-Solomon codes

A t error correcting R-S code with symbolsfrom GF(q) has the following parameters

Block length: n = q-1No. of parity-check digits: n-k = 2t

Minimum Distance: = 2t+1

ie.The length of the code is one less thanthe size of code symbols and the minumdistance is one greater than the number of

parity-check digitsWe generally use R-S codes with codesymbols from GF(2m). (ie. q =2m).


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Generator Polynomial

The minimal polynomial over GF(q) of an element in the same field GF(q) is

For a t error correcting R-S Code, the generatorpolynomial is

Any value of j0 can be used for an R-S code butj0=1 is conventional. Some clever choices can bemade for easy circuit implementations.

( )f x x =

0 0 01 2 1( ) ( )( )........( )j j j tg x x x x + + =


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The degree of the generator polynomial isalways 2t. Hence a Reed-Solomon code

satisfies

n k=2t


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Construction of g(x): Examples

(i) n=15, t=2

Since t=2, n-k=2x2=4. So this is a (15,11)code.

The coefficients of generator polynomial will

be elements from GF(16).Letj0=1.

2 3 4

4 3 2 3 3 2 2 3 2

4 13 3 6 2 3 10

( ) ( )( )( )( )( 1) ( ) ( 1)

g x x x x xx z z x z z x z x z z

x x x x

=

= + + + + + + + + +

= + + + +

The field elements of GF(16) are expressed as polynomials in z


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An information polynomial is a sequence ofeleven 16-ary(hexadecimal) symbols

(equivalent to 44 bits).(ii) n=7, t=2

This is R-S code over GF(8)Let j0= 4 4 5 6 0

4 2 3 2 2

4 6 3 6 2 3

( ) ( )( )( )( )

( 1) ( 1) ( 1)

g x x x x x

x z x z x z x z

x x x x

=

= + + + + + + +

= + + + +

The field elements of GF(8) are expressed as polynomials in z


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Here n k =2t=4 .Hence k=3.Aninformation polynomial is a sequence of

three 8-ary(octal) symbols (equivalent to9 bits).

Eg: Let the information polynomial be

Then the non-systematic codeword is

2 2( ) ( ) ( 1)m x z z x x z= + + + +


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4 2 3 4 6 3 6 2 3

4 6 5 6 4 3 2 5 4

( ) ( ) ( )

( )( )

0 0

c x m x g x

x x x x x x

x x x x x x

=

= + + + + + +

= + + + + + +

This is a sequence of seven octal symbols


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Special feature

Reed-Solomon code is a Maximum-Distance code.

The minimum distance dmin= n k+1

ie. R-S code satisfies the Singleton

bound with equality.This means that for a fixed (n,k), nocode can have a larger minimum

distance than a Reed-Solomoncode.


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16.reed solomon codes

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