A necessary and sufficient condition for a stabilizer code having transversal $CNOT$ is that the code is a CSS code (see Theorem 11.5 here or the question here).
I know that a sufficient condition for a code having transversal $H$ is that it is a self-dual CCS code (see here)
Is this condition also necessary? If not, is there a necessary condition that captures a family of stabilizer codes having transversal $H$ (like for the $CNOT$ case).
To clarify: I am aware of the necessary conditions that the logical $H$ must be in the normalizer of the stabilizer and that in the code space, it transforms the logical $X$ and $Z$ into each other by conjugation.
