= Reed-Solomon key equation
{c}
{title2=$\Lambda(z)S(z)\equiv\Omega(z)\pmod{z^{n-k}}$}
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 $\Lambda(0)=1$.
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