What are the transversal gates of the perfect 5-qubit code ( [[5,1,3]] code) ?
Since this is a stabilizer code we have transversal Pauli gates.
Are there any others?
I am primarily interested in single qubit gates that are transversal for a single block of the code. But I would also be very curious to see any two qubit gates which are transversal for two blocks of the code (and higher for three blocks of the code etc... but I would imagine that is very hard to find).
I know that $ Z $ is transversal because $ |0_L> $ is a superposition of even parity bit strings and $ |1_L> $ is a superposition of odd parity bit strings.
However the phase gate $ S=\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} $ is not transversal since $ |0_L> $ is not doubly even. See Transversal logical gate for Stabilizer (or at least Steane code)