New Error Correcting Codes from Lifting Alan Xinyu Guo

121 Pages·2015·0.65 MB·English

New Error Correcting Codes from Lifting
by
Alan Xinyu Guo
B.S., Duke University (2011)
S.M., Massachusetts Institute of Technology (2013)
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
○c Massachusetts Institute of Technology 2015. All rights reserved.
Author....................................................................
Department of Electrical Engineering and Computer Science
May 1, 2015
Certified by ...............................................................
Madhu Sudan
Adjunct Professor
Thesis Supervisor
Accepted by...............................................................
Leslie A. Kolodziejski
Chair, Department Committee on Graduate Students
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New Error Correcting Codes from Lifting
by
Alan Xinyu Guo
Submitted to the Department of Electrical Engineering and Computer Science
on May 1, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Electrical Engineering and Computer Science
Abstract
Error correcting codes have been widely used for protecting information from noise. The
theory of error correcting codes studies the range of parameters achievable by such codes, as
well as the efficiency with which one can encode and decode them. In recent years, attention
has focused on the study of sublinear-time algorithms for various classical problems, such
as decoding and membership verification. This attention was driven in part by theoretical
developments in probabilistically checkable proofs (PCPs) and hardness of approximation.
Locally testable codes (codes for which membership can be verified using a sublinear number
of queries) form the combinatorial core of PCP constructions and thus play a central role
in computational complexity theory. Historically, low-degree polynomials (the Reed-Muller
code) have been the locally testable code of choice. Recently, “affine-invariant” codes have
come under focus as providing potential for new and improved codes.
Inthisthesis, weexploitanaturalalgebraicoperationknownas“lifting”toconstructnew
affine-invariantcodesfromshorterbasecodes. Theseliftedcodesgenericallypossessdesirable
combinatorial and algorithmic properties. The lifting operation preserves the distance of the
base code. Moreover, lifted codes are naturally locally decodable and testable. We tap
deeper into the potential of lifted codes by constructing the “lifted Reed-Solomon code”,
a supercode of the Reed-Muller code with the same error-correcting capabilities yet vastly
greater rate.
The lifted Reed-Solomon code is the first high-rate code known to be locally decodable up
to half the minimum distance, locally list-decodable up to the Johnson bound, and robustly
testable, with robustness that depends only on the distance of the code. In particular, it is
the first high-rate code known to be both locally decodable and locally testable. We also
apply the lifted Reed-Solomon code to obtain new bounds on the size of Nikodym sets, and
also to show that the Reed-Muller code is robustly testable for all field sizes and degrees up
to the field size, with robustness that depends only on the distance of the code.
Thesis Supervisor: Madhu Sudan
Title: Adjunct Professor
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Acknowledgments
First andforemost, Ithank myadvisor, MadhuSudan, for hispatient guidance, support, and
encouragement throughout my four years at MIT. Our shared taste in finding clean, general
solutions to algebraic problems has led to many collaborations. I thank Scott Aaronson and
Dana Moshkovitz for serving on my thesis committee.
I am also indebted to Ezra Miller, who guided my undergraduate research at Duke, and
to Vic Reiner and Dennis Stanton who mentored me at their REU in Minnesota.
I thank my co-authors during graduate school: Greg Aloupis, Andrea Campagna, Erik
Demaine, Elad Haramaty, Swastik Kopparty, Ronitt Rubinfeld, Madhu Sudan, and Giovanni
Viglietta. I especially thank Ronitt and Piotr Indyk for their guidance early on, Erik for his
excitement and willingness to entertain my more fun research ideas (i.e. hardness of video
games), and Swastik for hosting my visit to Rutgers and teaching me about codes and PCPs.
I also thank friends and faculty with whom I have shared conversations at MIT: Pablo
Azar, Mohammad Bavarian, Eric Blais, Adam Bouland, Mahdi Cheraghchi, Henry Cohn,
Matt Coudron, Michael Forbes, Badih Ghazi, Pritish Kamath, Ameya Velingker, Henry
Yuen, and so many others.
I thank my friends outside of the field for their friendship and the good times during the
past four years. I especially thank Vivek Bhattacharya, Sarah Freitas, Henry Hwang, Steven
Lin, Ann Liu, Lakshya Madhok, and Roger Que.
I am grateful to my family. Without my parents, I would not exist, and neither would
this thesis. Moreover, they always supported my education and encouraged me to pursue
my dreams. I thank my younger sister Julia for encouraging me to set a good example.
Finally, my biggest thanks goes to Lisa — my fianc´ee, best friend, and companion for the
rest of my life. Graduate school was not always easy, but you were always there to support
me and pick me up when I was down, and to share in my triumphs. Thank you for your
unwavering love and loyalty, without which I do not believe I would have survived through
graduate school. Only you know every twist and turn my journey has taken. Without
hesitation, I dedicate this thesis to you.
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Contents
1 Introduction 11
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.2 PCPs and Local Algorithms . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.3 Affine-Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2 This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.2 Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.3 Robust Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Preliminaries 27
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Probability and Concentration bounds . . . . . . . . . . . . . . . . . . . . . 29
3 Error Correcting Codes 31
3.1 Classical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Local decoding, correcting, and list-decoding . . . . . . . . . . . . . . . . . . 32
3.3 Local testing and robust testing . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.1 Reed-Solomon code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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3.4.2 Reed-Muller code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Affine-invariance 39
4.1 Affine-invariant Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Equivalence of Invariance under Affine Transformations and Permutations . . 40
5 Lifting Codes 47
5.1 The Lift Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Algebraic and Combinatorial Properties . . . . . . . . . . . . . . . . . . . . 48
5.2.1 Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.2 Distance of Lifted Codes . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Local Decoding and Correcting . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 Local Correcting up to 1/4 Distance . . . . . . . . . . . . . . . . . . 53
5.3.2 Local Correcting up to 1/2 Distance . . . . . . . . . . . . . . . . . . 54
5.3.3 Local Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Local Testing and Robust Testing . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.1 Local Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.2 Robust Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 Robust Testing of Lifted Codes 61
6.1 Robustness of Lifted Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.1.2 Robustness for Small Dimension . . . . . . . . . . . . . . . . . . . . . 62
6.1.3 Robustness of Special Tensor Codes . . . . . . . . . . . . . . . . . . . 66
6.1.4 Robustness for Large Dimension . . . . . . . . . . . . . . . . . . . . . 74
6.2 Technical Algebraic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.1 Degree Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2.2 Analysis of Subspace Restrictions . . . . . . . . . . . . . . . . . . . . 85
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7 Applications 91
7.1 Lifted Reed-Solomon Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.1.1 Relationship to Reed-Muller . . . . . . . . . . . . . . . . . . . . . . . 92
7.1.2 Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.1.3 Global List-Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1.4 Local List-Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.1.5 Main Result: The Code That Does It All . . . . . . . . . . . . . . . . 102
7.2 Robust Low-Degree Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.3 Nikodym Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A Algebra Background 107
A.1 Arithmetic over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2 Tensor codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B Finite field geometry 111
B.1 Affine maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.2 Affine subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
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