[Paper Review] On Higher-Order Fourier Analysis over Non-Prime Fields

Arnab Bhattacharyya, Abhishek Bhowmick|arXiv (Cornell University)|May 4, 2015

Coding theory and cryptography42 references3 citations

TL;DR

This paper extends higher-order Fourier analysis to non-prime finite fields, enabling new algorithmic and structural results in coding theory and complexity. It proves that the list decoding radius of generalized Reed-Muller codes over any finite field equals their minimum distance, gives a polynomial-time algorithm for polynomial decomposition over arbitrary finite fields, and establishes testability of locally characterized affine-invariant properties.

ABSTRACT

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas. * For any fixed finite field $\mathbb{K}$, we show that the list decoding radius of the generalized Reed Muller code over $\mathbb{K}$ equals the minimum distance of the code. Previously, this had been proved over prime fields [BL14] and for the case when $|\mathbb{K}|-1$ divides the order of the code [GKZ08]. * For any fixed finite field $\mathbb{K}$, we give a polynomial time algorithm to decide whether a given polynomial $P: \mathbb{K}^n o \mathbb{K}$ can be decomposed as a particular composition of lesser degree polynomials. This had been previously established over prime fields [Bha14, BHT15]. * For any fixed finite field $\mathbb{K}$, we prove that all locally characterized affine-invariant properties of functions $f: \mathbb{K}^n o \mathbb{K}$ are testable with one-sided error. The same result was known when $\mathbb{K}$ is prime [BFHHL13] and when the property is linear [KS08]. Moreover, we show that for any fixed finite field $\mathbb{F}$, an affine-invariant property of functions $f: \mathbb{K}^n o \mathbb{F}$, where $\mathbb{K}$ is a growing field extension over $\mathbb{F}$, is testable if it is locally characterized by constraints of bounded weight.

Motivation & Objective

• To extend higher-order Fourier analysis tools to general finite fields, not just prime fields.
• To resolve Conjecture 1.1 on the list decoding radius of generalized Reed-Muller codes over arbitrary finite fields.
• To provide a polynomial-time algorithm for decomposing polynomials over any finite field into compositions of lower-degree polynomials.
• To prove that all locally characterized affine-invariant properties over functions from K^n to K are testable with one-sided error, for any fixed finite field K.
• To extend the testability of affine-invariant properties to growing field extensions over a fixed base field, under bounded-weight local constraints.

Proposed method

• Adapts higher-order Fourier analysis tools—especially Gowers uniformity norms and regularity lemmas—for general finite fields, not just prime fields.
• Uses a recursive decomposition strategy based on hyperplane restrictions and rank preservation under restriction, proving that rank drops by at most q when restricted to a hyperplane.
• Applies a blackbox reduction from algorithmic list decoding to combinatorial list decoding, leveraging the new structural results on polynomial representations.
• Employs a syntactic regularity lemma via the algorithm of Bhowmick, Lovett, and Tulsiani to regularize functions into structured factors over non-prime fields.
• Introduces a recursive algorithm that reduces the number of variables by fixing a variable not appearing in any low-degree factor, then lifts the solution via trace maps and field embeddings.
• Uses the Schwartz-Zippel-type lemma for multivariate polynomials to show that if a function agrees with a structured form on a large set, it must be independent of certain variables, enabling structural recovery.

Experimental results

Research questions

• RQ1 Does the list decoding radius of generalized Reed-Muller codes over any finite field K equal their minimum distance, as previously shown only over prime fields?
• RQ2 Can polynomial decomposition into compositions of lower-degree polynomials be decided in polynomial time over arbitrary finite fields, not just prime fields?
• RQ3 Are all locally characterized affine-invariant properties of functions f: K^n → K testable with one-sided error, even when K is not a prime field?
• RQ4 Can the testability of affine-invariant properties be extended to the case where K is a growing field extension over a fixed prime field F, under bounded-weight local constraints?
• RQ5 How does the rank of a polynomial behave under restriction to a hyperplane in non-prime fields, and can this be used to maintain regularity in higher-order Fourier analysis?

Key findings

• The list decoding radius of the generalized Reed-Muller code RM_K(n,d) over any finite field K equals its minimum distance δ_K(d), resolving Conjecture 1.1.
• For any fixed finite field K, there exists a polynomial-time algorithm to decide whether a given polynomial P: K^n → K admits a (k,Δ,Γ)-decomposition into lower-degree polynomials.
• All locally characterized affine-invariant properties of functions f: K^n → K are testable with one-sided error, for any fixed finite field K.
• For any fixed finite field F, an affine-invariant property of functions f: K^n → F, where K is a growing field extension over F, is testable if it is locally characterized by constraints of bounded weight.
• The rank of a polynomial over a non-prime field drops by at most q when restricted to a hyperplane, enabling recursive analysis in higher-order Fourier decomposition.