Noisy Interpolating Sets for low degree polynomials

Zeev Dvir*, Amir Shpilka

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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,..., 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.

Original languageEnglish
Title of host publicationProceedings - 23rd Annual IEEE Conference on Computational Complexity, CCC 2008
Pages140-148
Number of pages9
DOIs
StatePublished - 2008
Externally publishedYes
Event23rd Annual IEEE Conference on Computational Complexity, CCC 2008 - College Park, MD, United States
Duration: 23 Jun 200826 Jun 2008

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Conference

Conference23rd Annual IEEE Conference on Computational Complexity, CCC 2008
Country/TerritoryUnited States
CityCollege Park, MD
Period23/06/0826/06/08

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