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 $e(X)=a{X}^j$, the evaluation $r(\alpha )=a{\alpha }^j$ identifies its value and position when the relevant powers of $\alpha$ are distinct.

For a binary cyclic code with defining zeros of a cyclic code $\alpha$ and ${\alpha }^3$, a single error in coordinate $j$ has polynomial $X^j$ and syndromes $S_1={\alpha }^j$, $S_3={\alpha }^{3j}$. If $S_1\ne 0$ and $S_3=S_1^3$, recover $j$ from the exponent of $S_1$, flip that coordinate, and verify that the resulting polynomial vanishes at both roots. When the linear code has minimum Hamming distance at least five, this verified correction is the unique codeword within two errors. The syndrome relation alone should not replace checking the corrected word.