NDLI: Approximate List-Decoding of Direct Product Codes and Uniform Hardness Amplification

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Jaiswal, Ragesh Impagliazzo, Russell Kabanets, Valentine
Copyright Year	2009
Abstract	Given a message $msg\in\{0,1\}^N$, its k-wise direct product encoding is the sequence of k-tuples $(msg(i_1),\dots,msg(i_k))$ over all possible k-tuples of indices $(i_1,\dots,i_k)\in\{1,\dots,N\}^k$. We give an efficient randomized algorithm for approximate local list-decoding of direct product codes. That is, given oracle access to a word which agrees with a k-wise direct product encoding of some message $msg\in\{0,1\}^N$ in at least $\epsilon\geqslant{poly}(1/k)$ fraction of positions, our algorithm outputs a list of ${poly}(1/\epsilon)$ strings that contains at least one string $msg'$ which is equal to $msg$ in all but at most $k^{-\Omega(1)}$ fraction of positions. The decoding is local in that our algorithm outputs a list of Boolean circuits so that the jth bit of the ith output string can be computed by running the ith circuit on input j. The running time of the algorithm is polynomial in $\log N$ and $1/\epsilon$. In general, when $\epsilon>e^{-k^{\alpha}}$ for a sufficiently small constant $\alpha>0$, we get a randomized approximate list-decoding algorithm that runs in time quasi-polynomial in $1/\epsilon$, i.e., $(1/\epsilon)^{{poly}\log1/\epsilon}$. As an application of our decoding algorithm, we get uniform hardness amplification for ${P}^{{NP}_{\parallel}}$, the class of languages reducible to ${NP}$ through one round of parallel oracle queries: If there is a language in ${P}^{{NP}_{\parallel}}$ that cannot be decided by any ${BPP}$ algorithm on more than $1-1/n^{\Omega(1)}$ fraction of inputs, then there is another language in ${P}^{{NP}_{\parallel}}$ that cannot be decided by any ${BPP}$ algorithm on more than $1/2+1/n^{\omega(1)}$ fraction of inputs.
Starting Page	564
Ending Page	605
Page Count	42
File Format	PDF
ISSN	00975397
DOI	10.1137/070683994
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2 (Special Section on Foundations of Computer Science)
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-07-23
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Coding and information theory uniform hardness amplification error-correcting codes Analysis of algorithms and problem complexity approximately list-decodable codes direct product theorems Yao's XOR lemma Complexity classes
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