NDLI: Quadratic Goldreich--Levin Theorems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Tulsiani, Madhur Wolf, Julia
Copyright Year	2014
Abstract	Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich--Levin algorithm [O. Goldreich and L. Levin, A Hard-Core Predicate for All One-Way Functions," in Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 25--32] can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients. In the study of quadratic Fourier analysis, higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part is correspondingly weaker. For example, it has been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the $U^3$ norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition. A key part of the algorithm is a local self-correction procedure for Reed--Muller codes of order 2 (over $\F_2^n$) for a function at distance $1/2-\e$ from a codeword. Given a function $f:\F_2^n \to \{-1,1\}$ at fractional Hamming distance $1/2-\e$ from a quadratic phase (which is a codeword of the Reed--Muller code of order 2), we give an algorithm that runs in time polynomial in $n$ and finds a codeword at distance at most $1/2-\eta$ for $\eta = \eta(\e)$. This is an algorithmic analogue of Samorodnitsky's result [A. Samorodnitsky, Low-Degree Tests at Large Distances, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing, 2007, pp. 506--515], which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the list-decoding radius. For this purpose, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers $U^3$ norm over $\F_2^n$.
Starting Page	730
Ending Page	766
Page Count	37
File Format	PDF
ISSN	00975397
DOI	10.1137/12086827X
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2 (Special Section on the Fifty-Second IEEE Annual Symposium on Foundations of Computer Science (FOCS 2011))
Volume Number	43
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2014-04-29
Access Restriction	Subscribed
Subject Keyword	decomposition theorems quadratic Fourier analysis Arithmetic combinatorics; higher degree uniformity additive combinatorics Reed--Muller codes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Quadratic Goldreich-Levin theorems

Quadratic Goldreich-Levin Theorems

Quadratic Goldreich-Levin Theorems

List decoding of Reed-Muller codes (2004)

A Fourier-Analytic Approach to Reed-Muller Decoding

Quadratic Goldreich-Levin theorems (2011)

The weight spectrum of two families of Reed-Muller codes

Repairing Generalized Reed-Muller Codes

Efficient multivariate low-degree tests via interactive oracle proofs of proximity for polynomial codes

Quadratic Goldreich--Levin Theorems

Similar Documents

Quadratic Goldreich-Levin theorems

Quadratic Goldreich-Levin Theorems

Quadratic Goldreich-Levin Theorems

List decoding of Reed-Muller codes (2004)

A Fourier-Analytic Approach to Reed-Muller Decoding

Quadratic Goldreich-Levin theorems (2011)

The weight spectrum of two families of Reed-Muller codes

Repairing Generalized Reed-Muller Codes

Efficient multivariate low-degree tests via interactive oracle proofs of proximity for polynomial codes

Quadratic Goldreich--Levin Theorems