 Self-dual code

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 $A{A}^T=I$.