Abstract
$\newcommand{\F}{{\mathbb F}}$ 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 obtain an explicit family of punctured Reed-Muller codes (codes that are restrictions of a Reed-Muller code to a subset of the coordinates) which are good codes and have an efficient decoding algorithm against a constant fraction of errors. To the best of our knowledge, even the existence of punctured Reed-Muller codes that are good codes was not previously known.
Highlights
An interpolating set for degree-d polynomials is a set of points S ⊆ Fn such that if q(x1, . . . , xn) is a degree-d polynomial over F and we are given the set of values (q(s))s∈S we can reconstruct q
A set S is a noisy interpolating set for degree-d polynomials if we can reconstruct q even if an adversary corrupts an ε fraction of the values in (q(s))s∈S
It is not difficult to prove that a random set of size O(nd) is a noisy interpolating set
Summary
An interpolating set for degree-d polynomials is a set of points S ⊆ Fn such that if q(x1, . . . , xn) is a degree-d polynomial over F and we are given the set of values (q(s))s∈S we can reconstruct q. In this paper we study the problem of giving an explicit construction of a small noisy interpolating set for degree-d polynomials that has an efficient interpolation algorithm against a constant fraction of errors. Our noisy interpolating sets are of size O(nd) and they give a construction of good punctured Reed-Muller codes that can be decoded efficiently. From our work, Viola [14], building on the works of [6, 9], showed (over any field) that d · S is PRG for degree-d polynomials whenever S is PRG for degree-1 polynomials In particular this result implies that d · S is a noisy interpolating set. In the second variant (proper NIS) we do not allow such repetitions, and require S to be a simple set (and not a multiset) This is what allows us to get punctured Reed-Muller codes. For each of these types we prove a composition theorem, saying that the sumset of a NIS for degree-1 polynomials is a NIS for higher-degree polynomials. (The non-multiset version of this theorem requires an additional condition on the initial NIS.) We combine these theorems with known constructions of error-correcting codes to obtain our final results
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