= Syndrome decoding
{title2=$s=Hy^T=He^T$}
For a <linear code> with <parity-check matrix> $H$, write the received word as $y=c+e$, where $c$ is a <codeword> and $e$ an error pattern. Its <syndrome> depends only on the error because $Hc^T=0$. A syndrome decoder chooses a representative error $\widehat e$ with that syndrome and returns $y-\widehat e$. Choosing a minimum-<Hamming weight> representative gives <minimum-distance decoding>; within the guaranteed correction radius it recovers the unique transmitted <codeword>. Several representatives may tie outside that radius.
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