$$\begin{bmatrix} XZZXI\\IXZZX\\XIXZZ\\ZXIXZ\end{bmatrix}$$
The above is the generator of the Hamming code scheme [[5,1,3]].
I'd like to know step by step, how should I encode a single physical qubit $|\varphi\rangle = \alpha|0\rangle + \beta|1\rangle$ into 5.
EDIT: Also, consider a transversal $CNOT$ with target over the code. What is the Clifford transformation to make it logical?
