Recovering polynomials over finite fields from noisy character values

TL;DR

This paper presents polynomial time algorithms for recovering polynomials over finite fields from noisy character values, extending decoding capabilities for certain error-correcting codes and employing advanced algebraic techniques.

Contribution

It introduces the first polynomial time algorithms for decoding polynomials from noisy character values and for dual-BCH codes, using novel pseudopolynomial concepts and algebraic methods.

Findings

Efficient decoding of polynomials with errors over finite fields.

First polynomial time decoding algorithm for dual-BCH codes.

Development of pseudopolynomials with derivative properties.

Abstract

Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the values of $\chi \circ g$ 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.