Root-evaluation syndrome for a cyclic code
= Root-evaluation syndrome for a cyclic code
Let $\alpha$ be a root of the <generator polynomial of a cyclic code> $g(X)$. Every codeword $c(X)$ satisfies $c(\alpha)=0$, so a received word $r=c+e$ satisfies $r(\alpha)=e(\alpha)$. For a single-symbol error $e(X)=aX^j$, the evaluation $r(\alpha)=a\alpha^j$ identifies its value and position when the relevant powers of $\alpha$ are distinct.