Hi! Hope everyone is doing awesome!

Let's take 5-qubit code, for example, we have these two logical states:

$|0_L\rangle=\frac{1}{4}[|00000\rangle+|10010\rangle+|01001\rangle+|10100\rangle+|01010\rangle-|11011\rangle-|00110\rangle-|11000\rangle$

$\quad\quad\quad-|11101\rangle-|00011\rangle-|11110\rangle-|01111\rangle-|10001\rangle-|01100\rangle-|10111\rangle-|00101\rangle]$

$|1_L\rangle=\frac{1}{4}[|11111\rangle+|01101\rangle+|10110\rangle+|01011\rangle+|10101\rangle+|00100\rangle-|11001\rangle-|00111\rangle$

$\quad\quad\quad-|00010\rangle-|11100\rangle-|00001\rangle-|10000\rangle-|01110\rangle-|10011\rangle-|01000\rangle-|11010\rangle]

$

Stabilizers for 5-qubit code is given by:

$\langle XZZXI, IXZZX,XIXXZZ,ZXIXZ\rangle$

From the stabilizers and logical $X$ and $Z$, we can straightforwardly find the logical 0 and 1 from $I+S$ (identity + stabilizer). I'm wondering, how do we go the other way round, i.e., from the logical 0 and 1, is there a systematic way to work out the 4 stabilizers in this case?

Thank you in advance!! :))