Locally correctable codes from lifting Alan Guo MIT CSAIL Joint work with Swastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)

Slides:

Advertisements

Similar presentations

PRG for Low Degree Polynomials from AG-Codes Gil Cohen Joint work with Amnon Ta-Shma.

Advertisements

Multiplicity Codes Swastik Kopparty (Rutgers) (based on [K-Saraf-Yekhanin ’11], [K ‘12], [K ‘14])

Approximate List- Decoding and Hardness Amplification Valentine Kabanets (SFU) joint work with Russell Impagliazzo and Ragesh Jaiswal (UCSD)

Of 24 11/20/2012TIFR: Deterministic Communication Amid Uncertainty1 ( Deterministic ) Communication amid Uncertainty Madhu Sudan Microsoft, New England.

Information and Coding Theory

List decoding Reed-Muller codes up to minimal distance: Structure and pseudo- randomness in coding theory Abhishek Bhowmick (UT Austin) Shachar Lovett.

Digital Codes Coding Gain For a given Bit error Rate, Coding Gain is defined as the increase required in C/N such that un-coded data yields the same BER.

$100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300.

Universal Communication Brendan Juba (MIT) With: Madhu Sudan (MIT)

Locally Decodable Codes

Rank Bounds for Design Matrices and Applications Shubhangi Saraf Rutgers University Based on joint works with Albert Ai, Zeev Dvir, Avi Wigderson.

Probabilistically Checkable Proofs (and inapproximability) Irit Dinur, Weizmann open day, May 1 st 2009.

Of 14 01/03/2015ISCA-2015: Reliable Meaningful Communication1 Reliable Meaningful Communication Madhu Sudan Microsoft, Cambridge, USA.

Sparse Random Linear Codes are Locally Decodable and Testable Tali Kaufman (MIT) Joint work with Madhu Sudan (MIT)

Chapter 3 Coding Theory.

CS151 Complexity Theory Lecture 9 April 27, 2004.

Of 29 12/09/2014 Invariance in Property 1 Two Decades of Property Testing Madhu Sudan Microsoft Research TexPoint fonts used in EMF. Read.

Of 30 Locality in Coding Theory Madhu Sudan MSR June 16, 2015ISIT: Locality in Coding Theory1.

Of 28 Probabilistically Checkable Proofs Madhu Sudan Microsoft Research June 11, 2015TIFR: Probabilistically Checkable Proofs1.

Of 29 August 4, 2015SIAM AAG: Algebraic Codes and Invariance1 Algebraic Codes and Invariance Madhu Sudan Microsoft Research.

$100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300.

Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes Kenji Yasunaga * Toru Fujiwara + * Kwansei Gakuin University, Japan +

Reed Solomon Code Doug Young Suh Last updated : Aug 1, 2009.

1 Yuan Luo Xi’an Jan Optimum Distance Profiles of Linear Block Codes Shanghai Jiao Tong University.

How to Construct a Correct and Scalable iBGP Configuration Mythili Vutukuru Joint work with Paul Valiant, Swastik Kopparty and Hari Balakrishnan.

On Locally Decodable Codes Self Correctable Codes t-private PIR and Omer Barkol, Yuval Ishai and Enav Weinreb Technion, Israel.

Of 22 10/07/2015UMass: Uncertain Communication1 Communication Amid Uncertainty Madhu Sudan Microsoft Research Based on Juba, S. (STOC 2008, ITCS 2011)

1 Asymptotically good binary code with efficient encoding & Justesen code Tomer Levinboim Error Correcting Codes Seminar (2008)

Reed-Solomon Codes Probability of Not Decoding a Symbol Correctly By: G. Grizzard North Carolina State University Advising Professor: Dr. J. Komo Clemson.

Network RS Codes for Efﬁcient Network Adversary Localization Sidharth Jaggi Minghua Chen Hongyi Yao.

Fidelity of a Quantum ARQ Protocol Alexei Ashikhmin Bell Labs  Classical Automatic Repeat Request (ARQ) Protocol  Quantum Automatic Repeat Request (ARQ)

ELT Workshop. Correction Techniques Mistakes and errors are bound to occur in any learning environment. Learners should be made aware of their mistakes.

RS – Reed Solomon Error correcting code. Error-correcting codes are clever ways of representing data so that one can recover the original information.

8.1 Determine whether the following statements are correct or not

296.3:Algorithms in the Real World

Tali Kaufman (Bar-Ilan)

Algebraic Property Testing:

On-Chip ECC for Low-Power SRAM Design

Umans Complexity Theory Lectures

Locality in Coding Theory

Error control coding for wireless communication technologies

Error-Correcting Codes:

Yahoo Mail Customer Support Number

Most Effective Techniques to Park your Manual Transmission Car

How do Power Car Windows Ensure Occupants Safety

What Are Preservatives And Additives.

Local Decoding and Testing Polynomials over Grids

Algebraic Codes and Invariance

Maximally Recoverable Local Reconstruction Codes

Local Error-Detection and Error-correction

RS – Reed Solomon List Decoding.

Locally Decodable Codes from Lifting

Invariance in Property Testing

Beer Therapy Madhu Ralf Vote early, vote often

Algebraic Property Testing

Low-Degree Testing Madhu Sudan MSR Survey … based on many works

Imperfectly Shared Randomness

Unequal Error Protection for Video Transmission over Wireless Channels

Multiplying Polynomials

Question: Think Talk Idea Talk Idea Talk Idea Notes: We Understand.

(Marvell Semiconductor, Inc.)

Types of Errors And Error Analysis.

Zeev Dvir (Princeton) Shachar Lovett (IAS)

Information security engineering laboratory

Presentation transcript:

Locally correctable codes from lifting Alan Guo MIT CSAIL Joint work with Swastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework of lifting codes – New lower bounds for Nikodym sets

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework of lifting codes – New lower bounds for Nikodym sets

Error correcting codes

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

Locality

Bivariate polynomial codes

Local correctability

A brief history of LCCs

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

Our contributions

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

Main idea

Dimension of lifted RS code

Reed-Muller Lifted Reed-Solomon

Dimension of lifted RS code Reed-Muller Lifted Reed-Solomon

Dimension of lifted RS code Reed-Muller Lifted Reed-Solomon

Dimension of lifted RS code Reed-Muller Lifted Reed-Solomon

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

General results

Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

Application to Nikodym sets

Summary Lifting – Natural operation – Build longer codes from short ones – Preserve distance – Gain locality – Can get high rate Applications outside of coding theory – Improve polynomial method (e.g. Nikodym sets)

Thank you!