Cyclotomic finite-field Fourier spectra: Galois descent, native subfields, and residual coding
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arXiv
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| Natura: | Preprint |
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2026
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| author | Kumallagov, David Sizikov, Daniil Zarubin, Anton |
| author_facet | Kumallagov, David Sizikov, Daniil Zarubin, Anton |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20062 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cyclotomic finite-field Fourier spectra: Galois descent, native subfields, and residual coding Kumallagov, David Sizikov, Daniil Zarubin, Anton Commutative Algebra Information Theory Group Theory 12E20, 11T71, 94B05, 94B15, 68W30 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. |
| title | Cyclotomic finite-field Fourier spectra: Galois descent, native subfields, and residual coding |
| topic | Commutative Algebra Information Theory Group Theory 12E20, 11T71, 94B05, 94B15, 68W30 |
| url | https://arxiv.org/abs/2605.20062 |