\section{Parity-Check Matrix Decoding}

\begin{lemma}[Orthogonality Relation]
	Let $C$ be a systematic $(n,k)$ linear code over $\F$ with a standard generator matrix $G$ and parity-check matrix $H$. Then, for any vector $v$ in $\F^n$, we have $vH=0$ (the zero vector) if and only if $v$ belongs to $C$.
\end{lemma}

\begin{theorem}[Parity-Check Matrix Decoding]
	Parity-check matrix decoding will correct any single error if and only if the rows of the parity-check matrix are nonzero and no one row is a scalar multiple of any other row.
\end{theorem}