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Recovering polynomials over finite fields from noisy character values

Swastik Kopparty

TL;DR

The paper develops polynomial-time decoders for recovering a polynomial $g(X)$ over a finite field from noisy evaluations of the quadratic residue character $\chi\circ g$, provided $\deg(g) \le \epsilon\sqrt{q}$ and the noise is a constant fraction of the $q$ samples. It extends the approach to additive characters in characteristic $2$, yielding efficient decoding of dual-BCH codes from a constant fraction of errors. Central to the methods are Stepanov-style interpolation and the Berlekamp–Welch framework, augmented by the novel concept of pseudopolynomials, which behave like low-degree objects on $\mathbb{F}_q$ while maintaining high degree globally. The work connects algorithmic decoding to Weil-type bounds, providing algorithmic versions of these fundamental number-theoretic estimates. Together, these results push forward the boundary between algebraic-geometry-inspired bounds and practical decoding, with potential extensions to broader classes of characters and projection problems.

Abstract

Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $χ$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the values of $χ\circ g$ on ${\mathbb F}_q$, with up to a constant fraction of the values having errors. This was previously unknown even for the case of no errors. We give a similar algorithm for additive characters of polynomials over fields of characteristic $2$. This gives the first polynomial time algorithm for decoding dual-BCH codes of polynomial dimension from a constant fraction of errors. Our algorithms use ideas from Stepanov's polynomial method proof of the classical Weil bounds on character sums, as well as from the Berlekamp-Welch decoding algorithm for Reed-Solomon codes. A crucial role is played by what we call *pseudopolynomials*: high degree polynomials, all of whose derivatives behave like low degree polynomials on ${\mathbb F}_q$. Both these results can be viewed as algorithmic versions of the Weil bounds for this setting.

Recovering polynomials over finite fields from noisy character values

TL;DR

The paper develops polynomial-time decoders for recovering a polynomial over a finite field from noisy evaluations of the quadratic residue character , provided and the noise is a constant fraction of the samples. It extends the approach to additive characters in characteristic , yielding efficient decoding of dual-BCH codes from a constant fraction of errors. Central to the methods are Stepanov-style interpolation and the Berlekamp–Welch framework, augmented by the novel concept of pseudopolynomials, which behave like low-degree objects on while maintaining high degree globally. The work connects algorithmic decoding to Weil-type bounds, providing algorithmic versions of these fundamental number-theoretic estimates. Together, these results push forward the boundary between algebraic-geometry-inspired bounds and practical decoding, with potential extensions to broader classes of characters and projection problems.

Abstract

Let be a polynomial over a finite field with degree , and let be the quadratic residue character. We give a polynomial time algorithm to recover (up to perfect square factors) given the values of on , with up to a constant fraction of the values having errors. This was previously unknown even for the case of no errors. We give a similar algorithm for additive characters of polynomials over fields of characteristic . This gives the first polynomial time algorithm for decoding dual-BCH codes of polynomial dimension from a constant fraction of errors. Our algorithms use ideas from Stepanov's polynomial method proof of the classical Weil bounds on character sums, as well as from the Berlekamp-Welch decoding algorithm for Reed-Solomon codes. A crucial role is played by what we call *pseudopolynomials*: high degree polynomials, all of whose derivatives behave like low degree polynomials on . Both these results can be viewed as algorithmic versions of the Weil bounds for this setting.
Paper Structure (25 sections, 10 theorems, 147 equations)

This paper contains 25 sections, 10 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Techniques
  4. Questions
  5. Preliminaries
  6. Polynomials, Hasse Derivatives and Multiplicities
  7. The quadratic residue character over fields of odd characteristic
  8. Additive characters in fields characteristic $2$
  9. $\chi \circ g$
  10. Correctness of the Algorithm
  11. A remark
  12. $\mathsf{Tr} \circ g$
  13. Correctness of the Algorithm
  14. A remark
  15. Theory of Pseudopolynomials
  16. ...and 10 more sections

Key Result

Lemma 2.3

Let $A(X), B(X) \in \mathbb{F}[X]$, and $\alpha, \beta \in \mathbb{F}$. Then for all $\ell \geq 0$:

Theorems & Definitions (24)

  • Definition 2.1: Hasse Derivatives
  • Definition 2.2: Multiplicity
  • Lemma 2.3: Hasse derivative rules
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 14 more