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7/28/2019 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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