Error evaluator polynomial ($\Omega (z)$)

For errors $E_i$ at distinct nonzero locations $X_i$, the evaluator associated with syndromes $S_{\ell }={\sum }_i{E}_i{X}_i^{b+\ell }$ is $\Omega (z)={\sum }_i{E}_i{X}_i^b{\prod }_{h\ne i}(1-X_{h}z)$. Its degree is smaller than the number of errors. Together with the error locator polynomial it gives the Reed-Solomon key equation and the Forney algorithm for error magnitudes.

For distinct error locations and the syndrome convention $S_{\ell }={\sum }_i{E}_i{X}_i^{b+\ell }$, the displayed formula recovers the error magnitudes from the error evaluator polynomial and error locator polynomial. At $z=X_i^{-1}$, the evaluator is $E_i{X}_i^b{\prod }_{h\ne i}(1-X_h/X_i)$, whereas the locator's formal derivative is $-X_i{\prod }_{h\ne i}(1-X_h/X_i)$. Their quotient gives the formula; the denominator is nonzero because the locations are distinct.

// Target: polynomial.bigb