Say $C$ is a linear additive code over $GF(4)$.
Let the dual code $C^{\perp}=${$u \in GF(4)|u \cdot \bar{c}=0, \forall c \in C$}.
Say if I know that every element of $C^{\perp}$ is orthogonal to itself and to other elements of $C^{\perp}$. For example $$u_{i}\cdot \bar{u_{j}}=0, \forall u_{i},u_{j} \in C^{\perp}.$$
Then how would I prove that $C$ is contained in $C^{\perp}$? I think it should be possible, but perhaps I am not making the criteria strict enough?
I wonder if it is something to do with the fact that $c \in C$ must be orthogonal (wrt Hermitian inner product) to every $c^{\perp} \in C^{\perp}$? And every $c^{\perp} \in C^{\perp}$ is self-orthogonal...but I can't really get it straight in my head.