Self-orthogonal binary subspace ($W\subseteq W^{\perp }$)

A linear subspace $W\subseteq {\mathbb{F}}_2^n$ is self-orthogonal if $u\cdot v=0$ for every $u,v\in W$. The standard bilinear form is nondegenerate, whence $\dim W+\dim W^{\perp }=n$ and $\dim W\le \lfloor n/2\rfloor$. The condition with $u=v$ means every vector has even Hamming weight. Such a binary linear code gives the linear algebra behind the Eventown theorem.

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.