A lot of well known codes (5 qubit code, 7 qubit Steane code, 9 qubit Shor code) have logical zero and local one which can be written as (a global scalar times) a linear combination of computational basis kets with only $ \pm 1 $ as coefficients.
Is it true that for any stabilizer code there always exists a choice of logical code words that can be written as (a global scalar times) a linear combination of computational basis kets where every coefficient is $ \pm 1, \pm i $?
Here is my reasoning for why this should be true :
Suppose that we have a stabilizer code with stabilizer $ S $. Then we can form a projector onto the codespace by $$ P=\sum_{g \in S} g $$ We can take the image of the computational basis under this projector $ P $. For any given $ g $ and any computational basis ket $ v $ then $$ gv=\pm v' $$ for some other computational basis ket $ v' $ (or possibly $ \pm i $ if $ g $ has some factors of $ Y $). Adding all this up then $ Pv $ must be some $ \mathbb{Z}[i] $ linear combination of the computational basis kets. Is there always a way to factor out a global scalar and just get $ \pm 1,\pm i $ relative coefficients?