Cyclotomic finite-field Fourier spectra: Galois descent, native subfields, and residual coding

Abstract

We develop a Galois descent approach to finite-field Fourier spectra over an arbitrary finite base field. Let $\mathbb K=\mathbb F_q$ and $\mathbb L=\mathbb F_{q^m}$. If a Fourier transform is applied to a $\mathbb K$-valued vector, then its spectrum is not an arbitrary element of $\mathbb L^n$: it satisfies the Frobenius consistency relation \[ V_s^q=V_{qs \bmod n}. \] We prove a general Galois-descent theorem for Fourier transforms on finite abelian groups, characterize the one-dimensional spectra as products of subfields indexed by $q$-cyclotomic classes, and show that the orbit-seed representation is optimal in base-field coordinates. For arbitrary vectors in $\mathbb L^n$, we study a two-stage representation $g=f+h$, where $f$ is class-consistent and $h$ is a residual. The residual optimization separates over cyclotomic classes. We give exact support minimization, a symbol weight enumerator for the class-consistent code, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds. We also discuss implementation consequences for trace decompositions, normal bases, canonical subfield embeddings, and sparse-polynomial residual backends.