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.

翻译：设 $g(X)$ 为有限域 ${\mathbb F}_q$ 上的多项式，其次数为 $o(q^{1/2})$，并令 $χ$ 为二次剩余特征。我们提出一种多项式时间算法，能够在 ${\mathbb F}_q$ 上给定 $χ\circ g$ 的值（其中至多恒定比例的值存在误差）的情况下，恢复 $g(X)$（精确到完全平方因子）。即使对于无误差的情况，这一问题先前也未知解法。对于特征为 $2$ 的域上的多项式加性特征，我们给出了类似的算法。这为从恒定比例误差中解码多项式维度的对偶BCH码提供了首个多项式时间算法。我们的算法借鉴了Stepanov多项式方法证明经典Weil特征和界的思想，以及Reed-Solomon码的Berlekamp-Welch解码算法。其中起关键作用的是我们称之为*伪多项式*的结构：这些是高次多项式，但其所有导数在 ${\mathbb F}_q$ 上均表现出低次多项式的特性。这两项结果均可视为该设定下Weil界的算法化版本。