List-decoding reed-muller codes over small fields
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Small Sample Spaces Cannot Fool Low Degree Polynomials
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
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A Noisy Interpolating Set (NIS) for degree $d$ polynomials is a set $S \subseteq \F^n$, where $\F$ is a finite field, such that any degree $d$ polynomial $q \in \F[x_1, \ldots,x_n]$ can be efficiently interpolated from its values on $S$, even if an adversary corrupts a constant fraction of the values. In this paper we construct explicit NIS for every prime field $\F_p$ and any degree $d$. Our sets are of size $O(n^d)$ and have efficient interpolation algorithms that can recover $q$ from a fraction $\exp(-O(d))$ of errors. Our construction is based on a theorem which roughly states that if $S$ is a NIS for degree 1 polynomials then $d \cdot S= \{ a_1 +\ldots + a_d \,|\, a_i \in S\}$ is a NIS for degree $d$ polynomials. Furthermore, given an efficient interpolation algorithm for $S$, we show how to use it in a black-box manner to build an efficient interpolation algorithm for $d \cdot S$. As a corollary we get an explicit family of punctured Reed-Muller codes that is a family of good codes that have an efficient decoding algorithm from a constant fraction of errors. To the best of our knowledge no such construction was known previously.