Editing BCH code (section)

===Calculate the syndromes===
The received vector <math>R</math> is the sum of the correct codeword <math>C</math> and an unknown error vector <math>E.</math> The syndrome values are formed by considering <math>R</math> as a polynomial and evaluating it at <math>\alpha^c, \ldots, \alpha^{c+d-2}.</math> Thus the syndromes are<ref>{{Harvnb|Lidl|Pilz|1999|p=229}}</ref>
:<math>s_j = R\left(\alpha^j\right) = C\left(\alpha^j\right) + E\left(\alpha^j\right)</math>

for <math>j = c</math> to <math>c + d - 2.</math>

Since <math>\alpha^{j}</math> are the zeros of <math>g(x),</math> of which <math>C(x)</math> is a multiple, <math>C\left(\alpha^j\right) = 0.</math> Examining the syndrome values thus isolates the error vector so one can begin to solve for it.

If there is no error, <math>s_j = 0</math> for all <math>j.</math> If the syndromes are all zero, then the decoding is done.