Editing Reed–Solomon error correction (section)

====Formulation====

The transmitted message, <math>(c_0, \ldots, c_i, \ldots,c_{n-1})</math>, is viewed as the coefficients of a polynomial
<math display="block">
 s(x) = \sum_{i=0}^{n-1} c_i x^i.
</math>

As a result of the Reed–Solomon encoding procedure, ''s''(''x'') is divisible by the generator polynomial
<math display="block">
 g(x) = \prod_{j=1}^{n-k} (x - \alpha^j),
</math>
where ''α'' is a primitive element.

Since ''s''(''x'') is a multiple of the generator ''g''(''x''), it follows that it "inherits" all its roots:
<math display="block">
 s(x) \bmod (x - \alpha^j) = g(x) \bmod (x - \alpha^j) = 0.
</math>
Therefore,
<math display="block">
 s(\alpha^j) = 0,\ j = 1, 2, \ldots, n - k.
</math>

The transmitted polynomial is corrupted in transit by an error polynomial
<math display="block">
 e(x) = \sum_{i=0}^{n-1} e_i x^i
</math>
to produce the received polynomial
<math display="block">
 r(x) = s(x) + e(x).
</math>

Coefficient ''e<sub>i</sub>'' will be zero if there is no error at that power of ''x'', and nonzero if there is an error. If there are ''ν'' errors at distinct powers ''i<sub>k</sub>'' of ''x'', then
<math display="block">
 e(x) = \sum_{k=1}^\nu e_{i_k} x^{i_k}.
</math>

The goal of the decoder is to find the number of errors (''ν''), the positions of the errors (''i<sub>k</sub>''), and the error values at those positions (''e<sub>i<sub>k</sub></sub>''). From those, ''e''(''x'') can be calculated and subtracted from ''r''(''x'') to get the originally sent message ''s''(''x'').