We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars $\mathsf{K}$ and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of $\mathsf{K}$ is $\ne 2$. Our algorithms return a list of valid sparse interpolants for the $N$ support points and run in polynomial-time. For standard power basis our algorithms sample at $N = \lfloor \frac{4}{3} E + 2 \rfloor B$ points, which are fewer points than $N = 2(E+1)B - 1$ given by Kaltofen and Pernet in 2014. For Chebyshev basis our algorithms sample at $N = \lfloor \frac{3}{2} E + 2 \rfloor B$ points, which are also fewer than the number of points required by the algorithm given by Arnold and Kaltofen in 2015, which has $N = 74 \lfloor \frac{E}{13} + 1 \rfloor$ for $B = 3$ and $E \ge 222$. Our method shows how to correct $2$ errors in a block of $4B$ points for standard basis and how to correct $1$ error in a block of $3B$ points for Chebyshev Basis.

中文翻译：

---

基于切比雪夫误差的稀疏插值超出冗余块解码

我们提出了稀疏插值算法，用于在 $\le E$ 评估可能是错误的情况下，从变量的不同值的 $N$ 评估中恢复具有 $\le B$ 项的多项式。我们的算法在标量 $\mathsf{K}$ 领域执行精确算术，项可以是变量或 Chebyshev 多项式的标准幂，在这种情况下 $\mathsf{K}$ 的特征是 $\ne 2$ . 我们的算法为 $N$ 支持点返回一个有效的稀疏插值列表，并在多项式时间内运行。对于标准幂基础，我们的算法在 $N = \lfloor \frac{4}{3} E + 2 \rfloor B$ 点处采样，这些点数少于 $N = 2(E+1)B - 1$ 给出的点Kaltofen 和 Pernet 在 2014 年。对于 Chebyshev 基础，我们的算法样本在 $N = \lfloor \frac{3}{2} E + 2 \rfloor B$ 点，这也少于 Arnold 和 Kaltofen 在 2015 年给出的算法所需的点数，其中 $N = 74 \lfloor \frac{E}{13} + 1 \rfloor$ for $B = 3$ 和 $ E \ge 222$。我们的方法展示了如何在标准基础上纠正 $4B$ 点块中的 $2$ 错误，以及如何在 Chebyshev 基础上纠正 $3B$ 点块中的 $1$ 错误。