Syndrome classification of code cosets
= Syndrome classification of code cosets
{title2=$\mathbb F_q^N/C\cong\mathbb F_q^{N-k}$}
For a rank-$(N-k)$ parity-check matrix of a linear code $C$, two words have equal syndromes exactly when their difference lies in $C$. Thus attainable syndromes label cosets bijectively. For a received word $c+e$, its syndrome is that of the error $e$, so minimum-weight representatives provide <minimum-distance decoding>.
// Target: algebra.bigb