Editing Reed–Solomon error correction (section)

====Error locators and error values====
<!-- There's confusion between index k and the k in (n,k). Literature also has confusion? Or does it use kappa? -->

For convenience, define the '''error locators''' ''X<sub>k</sub>'' and '''error values''' ''Y<sub>k</sub>'' as
<math display="block">
 X_k = \alpha^{i_k}, \quad Y_k = e_{i_k}.
</math>

Then the syndromes can be written in terms of these error locators and error values as
<math display="block">
 S_j = \sum_{k=1}^\nu Y_k X_k^j.
</math>

This definition of the syndrome values is equivalent to the previous since <math>{(\alpha^j)}^{i_k} = \alpha^{j \cdot i_k} = {(\alpha^{i_k})}^j = X_k^j</math>.

The syndromes give a system of {{math|''n'' − ''k'' ≥ 2''ν''}} equations in 2''ν'' unknowns, but that system of equations is nonlinear in the ''X<sub>k</sub>'' and does not have an obvious solution. However, if the ''X<sub>k</sub>'' were known (see below), then the syndrome equations provide a linear system of equations
<!--
Vandermonde comment. Matrix equation -->
<math display="block">\begin{bmatrix}
X_1^1 & X_2^1 & \cdots & X_\nu^1 \\
X_1^2 & X_2^2 & \cdots & X_\nu^2 \\
\vdots & \vdots & \ddots & \vdots \\
X_1^{n-k} & X_2^{n-k} & \cdots & X_\nu^{n-k} \\
\end{bmatrix}
\begin{bmatrix}
Y_1 \\ Y_2 \\ \vdots \\ Y_\nu
\end{bmatrix}
=
\begin{bmatrix}
S_1 \\ S_2 \\ \vdots \\ S_{n-k}
\end{bmatrix},
</math>
which can easily be solved for the ''Y<sub>k</sub>'' error values.

Consequently, the problem is finding the ''X<sub>k</sub>'', because then the leftmost matrix would be known, and both sides of the equation could be multiplied by its inverse, yielding Y''<sub>k</sub>''

In the variant of this algorithm where the locations of the errors are already known (when it is being used as an [[erasure code]]), this is the end. The error locations (''X<sub>k</sub>'') are already known by some other method (for example, in an FM transmission, the sections where the bitstream was unclear or overcome with interference are probabilistically determinable from frequency analysis). In this scenario, up to <math>n - k</math> errors can be corrected.

The rest of the algorithm serves to locate the errors and will require syndrome values up to <math>2\nu</math>, instead of just the <math>\nu</math> used thus far. This is why twice as many error-correcting symbols need to be added as can be corrected without knowing their locations.