If there is a binary vector space $\bar{S}$, which is contained in it's dual $\bar{S}^{\perp}$, does this mean that every element of $S$ must be self-orthogonal?
Let me explain the specific case (taken from "Quantum Error Correction Via Codes Over GF(4)", Calderbank, Rains, Shor, Sloane):
$\bar{E}$ is a $2n$-dimensional binary vector space. Its elements are written $(a|b)$. It is equipped with the inner product $((a|b),(a'|b'))= a.b' + a'.b$ (note this is a symplectic inner product).
$\bar{S}$ is an $(n-k)$-dimensional linear subspace of $\bar{E}$ which is contained in its dual $\bar{S}^{\perp}$, with respect to the inner product defined above.
So my understanding is that every element of $\bar{S}$ must be self orthogonal in order to satisfy the above criteria?