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

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

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

Decoding BCH code in general case

DECODING BCH CODE IN GENREALASE


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:


C[15,5]t=3c=(000000000000000)y=(000101000000100)

Example:

Roots: , ,Inverse of roots:

e=(000101000000100)


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.



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)]



4.Set the following initial conditions:

5.Execute the Euclidean algorithm for until a step ρ is reached for which:

(x)) 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:

(x)) 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)



set:

Since ,e=2&t=3 We execute the Euclidean algorithmuntil :



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

THANKS! For Your Attention

BCH CODE AND DECODING BCH

Engineering

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