Dual code ($C^{\perp }$)

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 $C^{\perp }$.

A linear code is self-dual when it equals its dual code under the standard dot product. For a length-$2k$ code with a rank-$k$ generator matrix $G$, this is equivalent to $G{G}^T=0$: orthogonality gives inclusion in the dual code, and equal dimensions give equality. Over ${\mathbb{F}}_2$, a generator matrix $(I\mid A)$ therefore generates a self-dual code exactly when $A{A}^T=I$.

For a binary self-dual code, every word $c$ satisfies $c\cdot c=0$. In characteristic two this is ${\sum }_i{c}_i=0$, so the all-ones word is orthogonal to every codeword and lies in the dual code, hence the code itself. The dual code dimension identity gives even length and half-length dimension.