= Odd-length binary self-orthogonal dual extension
{title2=$C^\perp=C\oplus\langle\mathbf1\rangle$}
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 $\mathbf1\in C^\perp$ but $\mathbf1\notin C$. Comparing <dimensions> proves the formula. The <dimension> hypothesis alone is insufficient: $C=\langle100\rangle$ at length three is a counterexample.
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