 Reed-Solomon key equation

With syndromes $S_{\ell }={\sum }_i{E}_i{X}_i^{b+\ell }$ and distinct error-location values $X_i$, form $S(z)={\sum }_{\ell =0}^{n-k-1}{S}_{\ell }{z}^{\ell }$ and the error locator polynomial $\Lambda (z)={\prod }_i(1-X_{i}z)$. There is an error evaluator polynomial $\Omega$ of degree less than the number of errors satisfying the key equation. When the number of errors is at most $\lfloor (n-k)/2\rfloor$, the Berlekamp-Massey algorithm or extended Euclidean algorithm recovers these polynomials, normalized by $\Lambda (0)=1$.