Single-error test from two binary BCH syndromes ($S_1={\alpha }^j,S_3={\alpha }^{3j}=S_1^3$)

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.