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Popular decoding algorithms are,
# Peterson Gorenstein Zierler algorithm
===Peterson Gorenstein Zierler Algorithm===
Petersons algorithm, is the step 2, of the generalized BCH decoding procedure. We use Peterson's algorithm, to calculate the error locator polynomial coefficients
<math> \lambda_1 , \lambda_2 ... \lambda_{2t} </math>
of a polynomial
<math> \Lambda(x) = 1 + \lambda_1 X + \lambda_2 X^2 + ... + \lambda_{2t}X^{2t} </math>.
Now the procedure of the Peterson Gorenstein Zierler algorithm, for a given <math>(n,k,d_{min}) </math> BCH code
<!-- # Berlekamp-Massey algorithm▼
designed to correct
<math>[t=\frac{d_{min}-1}{2}]</math>
errors, is
* First generate the Matrix of <math>2t</math> syndromes,
* Next generate the <math>S_{txt}</math> matrix with the elements, Syndrome values,
<math>S_{t \times t}=\begin{bmatrix}s_1&s_2&s_3&...&s_t\\
s_2&s_3&s_4...&s_{t+1}\\
s_3&s_4&s_5&...&s_{t+2}\\
...&...&...&...&...\\
s_{t+1}&s_{t+2}&s_{t+3}&...&s_{2t-1}\end{bmatrix}</math>
* Generate a <math>c_{tx1}</math> matrix with, elements,
<math>C_{t \times 1}=\begin{bmatrix}s_{t+1}\\
s_{t+2}\\
...\\
...\\
s_{2t}\end{bmatrix}
</math>
* Let <math>\Lambda</math> denote the unknown polynomial coefficients, which are given,using,
<math>\Lambda_{t \times 1} = \begin{bmatrix}\lambda_{1}\\
\lambda_{2}\\
\lambda_{3}\\
\lambda_{4}\\
...\\
\lambda_{t}\end{bmatrix}
</math>
* Form the matrix equation
<math>S_{t \times t} \Lambda_{t \times 1} = C_{t \times 1} </math>
* If the determinant of matrix <math>S_{t \times t}</math> exists, then we can actually, find an inverse of this matrix, and solve for the values of unknown <math>\Lambda</math> values.
* If <math> det(S_{t \times t}) = 0 </math>,
then
if <math> t = 0</math>
then
declare a empty error locator polynomial
stop peterson procedure.
end
set <math> t \rightarrow t -1</math>
continue from the beginning of petersons decoding
end
* After you have values of <math>\Lambda</math> you have with you the error locator polynomial.
* Stop peterson procedure.
▲<!-- # Berlekamp-Massey algorithm ==Berlekamp-Massey algorithm== -->
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