Does anyone know a better name than "weakly self-dual"/"self-orthogonal" for CSS codes where $H_X=H_Z$, for example the Steane code, and the color codes?
Details
In the discussion in the comments of Example CSS codes and the properties "doubly even" and "self dual" it was pointed out that (weakly) self-duality is the property of classical codes and not quantum codes and that maybe this naming should not catch on. I agree with the sentiment after I've been corrected by my colleague :) However, he did not have any good idea around how these codes should be called.
Classical coding terms based on https://errorcorrectionzoo.org/c/qubit_css and https://arxiv.org/pdf/quant-ph/9608006.pdf:
A code $C$ is self-orthogonal or weakly self-dual, when $C \subseteq C^\perp$. Dual-containing is when $C^\perp \subseteq C$.
Error Correction Zoo calls the $C_X=C_Z$ CSS codes "self-orthogonal" which is not much better:
To add to the confusing quality of this naming, all stabilizer codes map to classical codes over $GF(4)$ that are weakly self-dual under the symplectic product. So technically all stabilizer codes are self-orthogonal/weakly self-dual already!
