Odd-length binary self-orthogonal dual extension ($C^{\perp }=C\oplus ⟨1⟩$)

If a self-orthogonal binary subspace has odd length $n$ and dimension $(n-1)/2$, its dual code is obtained by adjoining the all-ones word. Self-orthogonality makes every codeword even-weight, so $1\in C^{\perp }$ but $1\notin C$. Comparing dimensions proves the formula. The dimension hypothesis alone is insufficient: $C=⟨100⟩$ at length three is a counterexample.