For physical $d$-dimensional qudits we can define $$X= \sum_{i=0}^{d-1} |i+1\rangle \langle i |$$ and $$Z = \sum_{i=0}^{d-1} \omega^i |i\rangle \langle i |,$$ with $\omega=e^{2\pi i/d}$. The Fourier gate $$F=\frac{1}{\sqrt{d}} \sum_{i,j=0}^{d-1} \omega^{ij} |i\rangle \langle j|$$ transforms one basis into the other, i.e. $F X F^\dagger = Z$.
Now consider a stabilizer error correction code. One can choose the logical $Z_L$ operator such that it commutes with all stabilizer generators, and one can choose $X_L$ such that it commutes with the stabilizer generators plus it fulfills $Z_L X_L = \omega X_L Z_L$. That should fix our logical qudit space. Now we can choose any operator that maps $X_L$ to $Z_L$ as our $F_L$, but is there any canonical way to construct it? I was hoping that there is some compact formula for $F_L$ in terms of $Z_L$ and $X_L$, but I couldn't find it.
To be clear, I am not asking for a decomposition of $F_L$ in terms of a circuit (even though that is interesting as well), but I am "only" looking for a canonical way to construct the unitary operator itself.