[Paper Review] On the Bias of Reed-Muller Codes over Odd Prime Fields

Paul Beame, Shayan Oveis Gharan+1 more|arXiv (Cornell University)|Jun 18, 2018

Coding theory and cryptography11 references3 citations

TL;DR

This paper establishes that random bounded-degree polynomials over odd prime fields exhibit exponentially small bias with high probability, generalizing prior results over 𝔽₂. The key contribution is a new extremal rank property for submatrices of Reed-Muller generator matrices, which enables exponential tail bounds on bias and weight distribution for Reed-Muller codes over odd prime fields.

ABSTRACT

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, such polynomials have exponentially small bias. This also yields an exponential tail bound on the weight distribution of Reed-Muller codes over odd prime fields. These results generalize bounds of Ben-Eliezer, Hod, and Lovett who proved similar results over $\mathbb{F}_2$. A key to our bounds is the proof of a new precise extremal property for the rank of sub-matrices of the generator matrices of Reed-Muller codes over odd prime fields. This extremal property is a substantial extension of an extremal property shown by Keevash and Sudakov for the case of $\mathbb{F}_2$. Our exponential tail bounds on the bias can be used to derive exponential lower bounds on the time for space-bounded learning of bounded-degree polynomials from their evaluations over odd prime fields.

Motivation & Objective

• To extend the known bias bounds for Reed-Muller codes from 𝔽₂ to odd prime fields.
• To establish that random bounded-degree polynomials over odd prime fields have exponentially small bias with high probability.
• To derive exponential tail bounds on the weight distribution of Reed-Muller codes over odd prime fields.
• To prove a new extremal rank property for submatrices of Reed-Muller generator matrices over odd prime fields, generalizing prior results over 𝔽₂.
• To apply the bias bounds to derive exponential lower bounds on time or space complexity for learning bounded-degree polynomials over odd prime fields.

Proposed method

• Leverages character sum techniques and exponential sums over finite fields to analyze the bias of multivariate polynomials over odd prime fields.
• Introduces a novel extremal rank property for submatrices of Reed-Muller generator matrices, extending a result by Keevash and Sudakov from 𝔽₂ to odd prime fields.
• Uses a recursive decomposition of polynomial coefficients and generator matrix submatrices to bound the bias via rank-based arguments.
• Applies Corollary 4.4 and Lemma 5.8 to compare and bound the difference in bias values across different coefficient configurations.
• Employs a case analysis on the structure of coefficient vectors to show that bias cannot increase under certain transformations, leading to the extremal rank property.
• Combines the extremal rank result with character sum concentration to derive exponential tail bounds on the bias of random polynomials.

Experimental results

Research questions

• RQ1 What is the typical bias of a random bounded-degree polynomial over an odd prime field 𝔽_p?
• RQ2 Can the exponential tail bounds on bias observed in Reed-Muller codes over 𝔽₂ be extended to odd prime fields?
• RQ3 What extremal property governs the rank of submatrices of Reed-Muller generator matrices over odd prime fields?
• RQ4 How does the bias of a Reed-Muller codeword relate to its weight distribution over odd prime fields?
• RQ5 What are the implications of bias concentration for space-bounded learning of bounded-degree polynomials over odd prime fields?

Key findings

• With probability exponentially close to 1, a random bounded-degree polynomial over an odd prime field 𝔽_p has bias at most 2^{-Ω(n)} for a constant depending on degree and field size.
• The fraction of Reed-Muller codewords with bias exceeding 2^{-c n} is at most 2^{-Ω(m)} for m = log |RM(d,n)|, generalizing the 𝔽₂ result of Ben-Eliezer et al.
• A new extremal rank property for submatrices of Reed-Muller generator matrices is proven, which holds over odd prime fields and extends the 𝔽₂ result of Keevash and Sudakov.
• The weight distribution of Reed-Muller codes over odd prime fields satisfies exponential tail bounds, implying that most codewords are close to balanced.
• The bias concentration result implies exponential lower bounds on time or space complexity for learning bounded-degree polynomials over odd prime fields from evaluations.
• The analysis shows that for p > 2, bounding |bias_j(f)| for all j ∈ 𝔽_p^* is necessary and sufficient to control the imbalance of a function f: 𝔽_p^n → 𝔽_p.