Are all $[[n, k, d]]$ quantum codes equivalent to additive self-orthogonal $GF(4)^n$ classical codes?

Theorem 2 of [1] states:

Suppose $C$ is an additive self-orthogonal sub-code of $\textrm{GF}(4)^n$, containing $2^{n-k}$ vectors, such that there are no vectors of weight $<d$ in $C^\perp/C$. Then any eigenspace of $\phi^{-1}(C)$ is an additive quantum-error-correcting code with parameters $[[n, k, d]]$.

where here $\phi: \mathbb{Z}_2^{2n} \rightarrow \textrm{GF}(4)^n$ is the map between the binary representation of $n$-fold Pauli operators and their associated codeword, and $C$ is self-orthogonal if $C \subseteq C^\perp$ where $C^\perp$ is the dual of $C$.

This tells us that each additive self-orthogonal $\textrm{GF}(4)^n$ classical code represents a $[[n, k, d]]$ quantum code.

My question is whether the reverse is also true, that is: is every $[[n, k, d]]$ quantum code represented by an additive self-orthogonal $\textrm{GF}(4)^n$ classical code?

Or equivalently: Are there any $[[n, k, d]]$ quantum codes that are not represented by an additive self-orthogonal $\textrm{GF}(4)^n$ classical code?

[1]: Calderbank, A. Robert, et al. "Quantum error correction via codes over GF (4)." IEEE Transactions on Information Theory 44.4 (1998): 1369-1387.

• Aren't the stabilizer codes such as Toric codes or color codes self orthogonal? there is an isomorphism between both!! – Tessaracter Jun 20 '18 at 18:08
• Sorry, I don't understand your point. I am looking for a quantum code that is not self-orthogonal, not examples of those that are. – SLesslyTall Jun 21 '18 at 7:42
• What is the question exactly? As far as I have understood in the question you are trying to find quantum codes that represent classical code? – Josu Etxezarreta Martinez Jun 21 '18 at 11:07
• No, I am trying to find out if all quantum codes (on qubits) have equivalent classical codes. For clarity, I have highlighted the exact question and added another rephrasing. – SLesslyTall Jun 21 '18 at 12:23

However, quantum codes can be constructed from non-self-orthogonal classical codes over $$GF(4)^n$$ by means of entanglement-assistance. In this constructions, an arbitrary classical code is selected, and by adding some Bell pairs in the qubit system, commutation between the stabilizers is obtained.