Small codes with transversal Hadamard

Question

The fewer qubits it uses the better. Also, it would be great if the code admitted a transversal logical $CNOT$.

Answer 1

Every CSS code has transversal CNOT gate. Moreover, if the $X$ and $Z$ sectors of the stabilizer are isomorphic then the code has transversal Hadamard. See this answer for proofs of both facts. The CSS construction is also described in section 10.4.2 on page 450 in Nielsen & Chuang.

This leads us to the smallest non-trivial code with transversal CNOT and Hadamard. Namely, the $[\![4,2,2]\!]$ code with stabilizer group generated by $XXXX$ and $ZZZZ$. See this question for arguments that rule out smaller codes.