= Single-error test from two binary BCH syndromes
{title2=$S_1=\alpha^j,\quad S_3=\alpha^{3j}=S_1^3$}
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 $S_1\ne0$ 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.
Back to article page
