Editing BCH code (section)

====Peterson–Gorenstein–Zierler algorithm====
<!-- this confuses t (max number of errors that can be corrected) with ν (actual number of errors) -->
Peterson's algorithm is the step 2 of the generalized BCH decoding procedure. Peterson's algorithm is used to calculate the error locator polynomial coefficients  <math> \lambda_1 , \lambda_2, \dots, \lambda_{v} </math> of a polynomial

: <math> \Lambda(x) = 1 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_v x^v .</math>

Now the procedure of the Peterson–Gorenstein–Zierler algorithm.<ref>{{Harvnb|Gorenstein|Peterson|Zierler|1960}}</ref> Expect we have at least 2''t'' syndromes ''s''<sub>''c''</sub>, …, ''s''<sub>''c''+2''t''−1</sub>. Let ''v''&nbsp;=&nbsp;''t''.
{{ordered list
| Start by generating the <math>S_{v\times v}</math> matrix with elements that are syndrome values
: <math>S_{v \times v}=\begin{bmatrix}s_c&s_{c+1}&\dots&s_{c+v-1}\\
s_{c+1}&s_{c+2}&\dots&s_{c+v}\\
\vdots&\vdots&\ddots&\vdots\\
s_{c+v-1}&s_{c+v}&\dots&s_{c+2v-2}\end{bmatrix}.
</math>
| Generate a <math>c_{v \times 1}</math> vector with elements
: <math>C_{v \times 1}=\begin{bmatrix}s_{c+v}\\
s_{c+v+1}\\
\vdots\\
s_{c+2v-1}\end{bmatrix}.
</math>
| Let <math>\Lambda</math> denote the unknown polynomial coefficients, which are given by
: <math>\Lambda_{v \times 1} = \begin{bmatrix}\lambda_{v}\\
\lambda_{v-1}\\
\vdots\\
\lambda_{1}\end{bmatrix}.
</math>
| Form the matrix equation
: <math>S_{v \times v} \Lambda_{v \times 1}  = -C_{v \times 1\,} .</math>
| If the determinant of matrix <math>S_{v \times v}</math> is nonzero, then we can actually find an inverse of this matrix and solve for the values of unknown <math>\Lambda</math> values.
| If <math>\det\left(S_{v \times v}\right) = 0,</math> then follow
        if <math>v = 0</math>
        then
              declare an empty error locator polynomial
              stop Peterson procedure.
        end
        set <math>v \leftarrow v -1</math>
        continue from the beginning of Peterson's decoding by making smaller <math>S_{v \times v}</math>
| After you have values of <math>\Lambda</math>, you have the error locator polynomial.
| Stop Peterson procedure.
}}