Noisy Interpolating Sets for low degree polynomials

A Noisy Interpolating Set (NIS) for degree d polynomials is a set S ⊆ double-struck F signⁿ, where F is a finite field, such that any degree d polynomial q ∈ F [x₁,..., 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.S = {a₁ +... + a_d | a_i ∈ 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.