This document presents information on BCH codes and their decoding. It discusses how BCH codes were invented in 1959-1960 and provides an abstract stating that BCH codes form a class of cyclic error-correcting codes constructed using finite fields. Various decoding methods for BCH codes are then described, including Chien search, the Euclidean algorithm, and the Berlekamp-Massey algorithm. The document goes on to explain the general decoding process for BCH codes and provides an example. It also covers correcting errors and erasures for nonbinary BCH codes using the Euclidean algorithm, demonstrating with an example.

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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
O
v
e
r
v
i
e
w

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DECODING BCH CODE IN
GENREALASE
Historical 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 algorithm
3. 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=3
c=(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 algorithm
until :

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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)

19. 1.F._J._MacWilliams,_N._J._A._Sloane. The Theory of
Error-Correcting Codes
2004-Error Control Coding-Lin&Castello.2
3.Steven Roman. Coding_and_information_theory

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