Frobenius orbit codes and finite-field Fourier spectra: exact distance and coset distributions

Let $q$ be a prime power, let $m\ge 1$, put $Q=q^m$, and let a permutation $\pi$ of a finite index set $I$ satisfy ${\pi }^m=1$. We study the $\text{F}q$-linear code $$\mathcal{C}(\pi )=\{x\in {\text{Lfield}}^I:x_{\pi (i)}=x_i^q\},$$ which includes the Fourier spectra of $\text{K}$-valued functions on finite abelian groups split by $\text{Lfield}$. Our structural starting point is a $\text{F}q$-linear Hamming isometry taking each cycle of $\pi$ of length $\mathrm{\ell }$ to the subfield repetition code $\{(b,\dots ,b):b\in \text{F}{q}^{\mathrm{\ell }}\}$. From this normal form we obtain exact symbol weight and received-word distance enumerators, all list sizes, and the covering radius. An occupancy generating function gives the complete coset-leader distribution, the exact mean distance, the probability of a unique nearest word, and the number of deep holes. For the full cyclic family of length $q^m-1$, the deficit of a uniformly random ambient word from the covering radius is asymptotically Poisson with mean $(m-1)\mathrm{/}2$, with an additional $1$ when $m$ is even; a central limit theorem follows. We also determine the deep-hole probability and the second-order gap to the sphere-covering bound. Classical Fourier–Galois descent and cyclotomic orbit enumeration enter only to identify the Fourier specialization.