 Parity-check matrix

For a linear code $C\subseteq {\mathbb{F}}_q^n$ of dimension $k$, a parity-check matrix is an $(n-k)\times n$ matrix $H$ whose rows form a basis of the dual code $C^{\perp }$. With codewords written as row vectors,Thus $H$ gives a full set of independent linear equations defining $C$. A received row vector $r$ has syndrome $H{r}^T$, which is zero exactly when $r$ is a codeword.

For example, $H=(111)$ defines the binary even-weight binary code of length $3$. It has rank $1$, so its kernel has dimension $2$. One can also use a check matrix with redundant rows; the same kernel is defined, but only an independent row selection is a generator matrix whose rows form a basis of $C^{\perp }$.