Download - BCH CODE AND DECODING BCH
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Presented by:Ahmad khosravani
DECODING BCH CODE
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Presented by:Ahmad khosravani
Historical of BCH
Decoding of binary BCH in general case
Abstract
Correction of errors and erasures for nonbinary BCH
Overview
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DECODING BCH CODE IN GENREALASEHistorical of BCH
BCH codes were invented in 1959 by
French mathematician
Alexis Hocquenghem,
and independently in 1960 byRaj
Chandra Boseand Dijen K. Ray-
Chaudhuri
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DECODING BCH CODE IN GENREALASE
Abstract
In coding theorey, the BCH codes form a class of cyclic error correcting code that are constructed using finite fields.
Various decoding for BCH code:1. Chien search 2. Euclidean algorithm3. the Berlekamp-Massey
Algorithm
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Decoding BCH code in general case
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DECODING BCH CODE IN GENREALASE
Decoding BCH code in general case
Let C be a nonbinary [n,k,d] code with designed distance odd. (i) Compute syndrome the
received vector y.
(ii) Compute the error locator polynomial.
(iii) Find the roots of error locator polynomial.
Decoding steps:
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Decoding BCH code in general case
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Decoding BCH code in general case
C[15,5]t=3c=(000000000000000)y=(000101000000100)
Example:
Roots: , ,Inverse of roots:
e=(000101000000100)
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Correction of errors and erasures for nonbinary BCH
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Correction of errors and erasures for nonbinary BCH
A q-ary t-error-correction BCH code can be used to correct all combinations of v symbols errors and e symbols erasures provided that the inequality
Holds.
In this section we let that erased position are known.
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Correction of errors and erasures for nonbinary BCH
Correction of errors and erasures for nonbinary BCH
Decoding prosess with Euclidean algorithm:
1.compute the erasure-location polynomial β(x).
2.Form the modified received polynomial by replaccing the erased symbols with zeros.Compute the syndromes polynomial s(x) from .
3.Compute the modified syndrome polynomial T(X)=[S(X) β(x)]
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Correction of errors and erasures for nonbinary BCH
Correction of errors and erasures for nonbinary BCH
4.Set the following initial conditions:
5.Execute the Euclidean algorithm for until a step ρ is reached for which:
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(x)) Correction of errors and erasures for nonbinary
BCH
Correction of errors and erasures for nonbinary BCH
6.Find the roots of σ(x) and determine the error location in r(x).
7.Determine the values of errors and erasure from and
The error values are given by:
And the value of erased symbols are given by:
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(x)) Correction of errors and erasures for nonbinary
BCH
Correction of errors and erasures for nonbinary BCH
Example:Consider the triple error correcting nonbinary BCH code of length 15 over GF( ) with:
V=2& e=2
e
c=(000000000000000)
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Correction of errors and erasures for nonbinary BCH
Correction of errors and erasures for nonbinary BCH
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Correction of errors and erasures for nonbinary BCH
Correction of errors and erasures for nonbinary BCH
set:
Since ,e=2&t=3 We execute the Euclidean algorithmuntil :
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Correction of errors and erasures for nonbinary BCH
Correction of errors and erasures for nonbinary BCH
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Correction of errors and erasures for nonbinary BCH
Correction of errors and erasures for nonbinary BCH
C(x)=e(x)+r(x)=(000000000000000)
Reference :
1.F._J._MacWilliams,_N._J._A._Sloane. The Theory ofError-Correcting Codes
2004-Error Control Coding-Lin&Castello . 2
3.Steven Roman. Coding_and_information_theory
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