I understood that to get fault-tolerant computing, one can start from any stabilizer group and then unitarily modifying it to get a new stabilizer group which relates to some logical operators.
For simplicity, this framework usually starts from a canonical stabilizer (which relates to an unencoded space). For example I read here how to get a code for each of the operator $H$, $CX$ and $\sqrt{Z}$. However, this means that I am creating a different code for each operation. To perform real computing, I'd imagine an encoding operation $U$, which gives a single code and, for which, $H$, $CX$ and $\sqrt{Z}$ are simultaneously transversal logical operator.
My question is then: what is such an encoding operator $U$?
