Editing Reed–Solomon error correction (section)

====Syndrome decoding====

The decoder starts by evaluating the polynomial as received at points <math>\alpha^1 \dots \alpha^{n-k}</math>. We call the results of that evaluation the "syndromes" ''S''<sub>''j''</sub>. They are defined as
<math display="block">
 \begin{align}
  S_j &= r(\alpha^j) = s(\alpha^j) + e(\alpha^j) = 0 + e(\alpha^j) \\
      &= e(\alpha^j) \\
      &= \sum_{k=1}^\nu e_{i_k} {(\alpha^j)}^{i_k}, \quad j = 1, 2, \ldots, n - k.
 \end{align}
</math>
Note that <math>s(\alpha^j) = 0</math> because <math>s(x)</math> has roots at <math>\alpha^j</math>, as shown in the previous section.

The advantage of looking at the syndromes is that the message polynomial drops out. In other words, the syndromes only relate to the error and are unaffected by the actual contents of the message being transmitted. If the syndromes are all zero, the algorithm stops here and reports that the message was not corrupted in transit.