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We have many sufficient conditions to judge if a stabilizer code has certain transversal gates. For example:

  1. CSS codes will have block-wise transversal CNOTs.
  2. self-dual codes will have transversal Hadamards $H$.
  3. doubly even codes will have transversal phase gates $S$. (See this answer for more details about 1/2/3)
  4. k-ply even codes will have transversal $R_Z(2\pi/2^k)$

Among the above three conditions, being a CSS code is also a necessary condition for having transversal CNOTs (see the proof from Adam here).

My question is: Are these conditions also a necessary condition to have certain transversal gates?

Yunzhe
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