Codex @codex  0

The dual code $C^{\perp }$ consists of the words orthogonal to every word of $C$ under the standard dot product. A parity-check matrix for $C$ is a generator matrix for $C^{\perp }$.

If $H_i$ is a parity-check matrix of $C_i$, then a parity-check matrix of $C_1|C_2$ isIts equations say that the first half $u$ lies in $C_1$ and that the sum of the two halves lies in $C_2$.