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

TL;DR

This paper introduces a Galois descent framework for finite-field Fourier spectra, characterizing spectra with Frobenius relations and optimizing residual representations for coding and computational applications.

Contribution

It develops a general Galois-descent theorem for Fourier transforms over finite fields, characterizes spectra via cyclotomic classes, and proposes optimized residual representations.

Findings

Spectra satisfy Frobenius consistency relations.

Exact support minimization and residual tail bounds are achieved.

Orbit-seed representation is proven optimal in base-field coordinates.

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.