This could be seen as a followup to the question "How to calculate the distance of stabilizer code?". Summarizing the accepted answer : distance is the minimum weight of the set $$E = \bigl\{e : e \not \in S, e \in \mathrm{Nor}(P_N,S)/(\pm I) \bigr\}$$ where $S$ is the stabilizer group (generated by $K_n$'s in the previous question), and $\mathrm{Nor}(P_N,S)$ is its normalizer in the Pauli group of order $2^{2N+1}$ (where $N$=number of qubits; using real version of group here).
My question is the following: does this hold for $k=0$ stabilizer codes? I suspect that it doesn't always hold but can't find a reference for it... it does seem to work for most cases, but some simple counter examples are also easy to find : take the GHZ state $\tfrac{1}{\sqrt 2}\bigl(\lvert00\rangle + \lvert11\rangle\bigr)$, with $K_1=X_1X_2$ and $K_2=Z_1Z_2$. In this case, $\mathrm{Nor}(P,S)=\pm S$, so the set $E$ is empty. Something is obviously broken in this process: I think that the distance should be 2. What's going on here?