Noisy Interpolating Sets for Low Degree Polynomials

A Noisy Interpolating Set (NIS) for degree d polynomials is a set S ⊆ F n, where F is a finite field, such that any degree d polynomial q ∈ F[x1,...,xn] 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 Fp 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 · S = {a1 +... + ad | ai ∈ 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 · 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