C Copyright 2007 Atri Rudra
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c Copyright 2007 Atri Rudra List Decoding and Property Testing of Error Correcting Codes Atri Rudra A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2007 Program Authorized to Offer Degree: Computer Science and Engineering University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by Atri Rudra and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of the Supervisory Committee: Venkatesan Guruswami Reading Committee: Paul Beame Venkatesan Guruswami Dan Suciu Date: In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.” Signature Date University of Washington Abstract List Decoding and Property Testing of Error Correcting Codes Atri Rudra Chair of the Supervisory Committee: Associate Professor Venkatesan Guruswami Department of Computer Science and Engineering Error correcting codes systematically introduce redundancy into data so that the original in- formation can be recovered when parts of the redundant data are corrupted. Error correcting codes are used ubiquitously in communication and data storage. The process of recovering the original information from corrupted data is called decod- ing. Given the limitations imposed by the amount of redundancy used by the error correct- ing code, an ideal decoder should efficiently recover from as many errors as information- theoretically possible. In this thesis, we consider two relaxations of the usual decoding procedure: list decoding and property testing. A list decoding algorithm is allowed to output a small list of possibilities for the original information that could result in the given corrupted data. This relaxation allows for effi- cient correction of significantly more errors than what is possible through usual decoding procedure which is always constrained to output the transmitted information. We present the first explicit error correcting codes along with efficient list-decoding • algorithms that can correct a number of errors that approaches the information-theoretic limit. This meets one of the central challenges in the theory of error correcting codes. We also present explicit codes defined over smaller symbols that can correct signifi- • cantly more errors using efficient list-decoding algorithms than existing codes, while using the same amount of redundancy. We prove that an existing algorithm for a specific code family called Reed-Solomon • codes is optimal for “list recovery,” a generalization of list decoding. Property testing of error correcting codes entails “spot checking” corrupted data to quickly determine if the data is very corrupted or has few errors. Such spot checkers are closely related to the beautiful theory of Probabilistically Checkable Proofs. We present spot checkers that only access a nearly optimal number of data symbols • for an important family of codes called Reed-Muller codes. Our results are the first for certain classes of such codes. We define a generalization of the “usual” testers for error correcting codes by en- • dowing them with the very natural property of “tolerance,” which allows slightly corrupted data to pass the test. TABLE OF CONTENTS Page ListofFigures...................................... vi ListofTables ...................................... vii Chapter1: Introduction.............................. 1 1.1 BasicsofErrorCorrectingCodes. ... 2 1.1.1 Historical Background and Modeling the Channel Noise ...... 3 1.2 ListDecoding................................. 5 1.2.1 GoingBeyondHalftheDistanceBound . 8 1.2.2 WhyisListDecodingAnyGood?. 10 1.2.3 The Challenge of List Decoding (and What Was Already Known) . 12 1.3 PropertyTestingofErrorCorrectingCodes . ...... 13 1.3.1 ABriefHistoryofPropertyTestingofCodes . ... 14 1.4 ContributionsofThisThesis . .. 15 1.4.1 ListDecoding............................. 15 1.4.2 PropertyTesting ........................... 17 1.4.3 OrganizationoftheThesis . 19 Chapter2: Preliminaries ............................. 20 2.1 TheBasics................................... 20 2.1.1 BasicDefinitionsforCodes . 21 2.1.2 CodeFamilies ............................ 22 2.1.3 LinearCodes ............................. 22 2.2 Preliminaries and Definitions Related to List Decoding . .......... 24 2.2.1 Ratevs.Listdecodability . 26 2.2.2 Results Related to the q-aryEntropyFunction . 30 2.3 DefinitionsRelatedtoPropertyTestingofCodes . ....... 34 i 2.4 CommonFamiliesofCodes . 35 2.4.1 Reed-SolomonCodes . 35 2.4.2 Reed-MullerCodes. 36 2.5 BasicFiniteFieldAlgebra . 36 Chapter3: ListDecodingofFoldedReed-SolomonCodes . ....... 38 3.1 Introduction.................................. 38 3.2 FoldedReed-SolomonCodes. 41 3.2.1 DescriptionofFoldedReed-SolomonCodes . .. 41 3.2.2 WhyMightFoldingHelp? . 43 3.2.3 RelationtoParvareshVardyCodes. 44 3.3 Problem Statement and Informal Description of the Algorithms. 45 3.4 TrivariateInterpolationBasedDecoding . ....... 48 3.4.1 FactsaboutTrivariateInterpolation . .... 49 3.4.2 Using Trivariate Interpolation for Folded RS Codes . ....... 51 3.4.3 Root-findingStep. .. .. .. .. .. .. .. 53 3.5 CodesApproachingListDecodingCapacity . ..... 56 3.6 ExtensiontoListRecovery . 63 3.7 BibliographicNotesandOpenQuestions. ..... 66 Chapter4: ResultsviaCodeConcatenation . ..... 69 4.1 Introduction.................................. 69 4.1.1 CodeConcatenationandListRecovery . 70 4.2 Capacity-Achieving Codes over Smaller Alphabets . ........ 72 4.3 BinaryCodesListDecodableuptotheZyablovBound . ...... 76 4.4 UniqueDecodingofaRandomEnsembleofBinary Codes . ..... 77 4.5 ListDecodinguptotheBlokhZyablovBound. .... 80 4.5.1 MultilevelConcatenatedCodes . 81 4.5.2 LinearCodeswithGoodNestedListDecodability . .... 84 4.5.3 ListDecodingMultilevelConcatenatedCodes . .... 88 4.5.4 PuttingitTogether . .. .. .. .. .. .. .. 92 4.6 BibliographicNotesandOpenQuestions. ..... 94 ii Chapter 5: List Decodability of Random Linear Concatenated Codes. 96 5.1 Introduction.................................. 96 5.2 Preliminaries ................................. 97 5.3 OverviewoftheProofTechniques . 101 5.4 ListDecodabilityofRandomConcatenatedCodes . .......103 5.5 UsingFoldedReed-SolomonCodeasOuterCode . .111 5.5.1 Preliminaries .............................111 5.5.2 TheMainResult ...........................113 5.6 BibliographicNotesandOpenQuestions. .117 Chapter6: LimitstoListDecodingReed-SolomonCodes . .......119 6.1 Introduction..................................119 6.2 OverviewoftheResults. 120 6.2.1 LimitationstoListRecovery . 120 6.2.2 Explicit“Bad”ListDecodingConfigurations . .122 6.2.3 ProofApproach. .. .. .. .. .. .. .. .123 6.3 BCH CodesandListRecoveringReed-SolomonCodes . .123 6.3.1 MainResult..............................123 6.3.2 ImplicationsforReed-SolomonListDecoding . .129 6.3.3 Implications for List Recovering Folded Reed-SolomonCodes . 130 6.3.4 A Precise Description of Polynomials with Values in BaseField . 131 6.3.5 SomeFurtherFactsonBCHCodes . 133 6.4 Explicit Hamming Balls with Several Reed-Solomon Codewords . 135 6.4.1 ExistenceofBadListDecodingConfigurations . .135 6.4.2 LowRateReed-SolomonCodes . 136 6.4.3 HighRateReed-SolomonCodes . 140 6.5 BibliographicNotesandOpenQuestions. .143 Chapter7: LocalTestingofReed-MullerCodes . ......147 7.1 Introduction..................................147 7.1.1 ConnectiontoCodingTheory . 148 7.1.2 OverviewofOurResults . .148 7.1.3 OverviewoftheAnalysis. .150 iii 7.2 Preliminaries .................................151 7.2.1 FactsfromFiniteFields . .154 7.3 Characterization of Low Degree Polynomials over Fp ............156 Fn 7.4 A Tester for Low Degree Polynomials over p ................162 7.4.1 Testerin Fp ..............................163 7.4.2 Analysis of Algorithm Test- ....................164 Pt 7.4.3 ProofofLemma7.11. .167 7.4.4 ProofofLemma7.12. .173 7.4.5 ProofofLemma7.13. .178 7.5 ALowerBoundandImprovedSelf-correction . .179 7.5.1 ALowerBound............................179 7.5.2 ImprovedSelf-correction. 180 7.6 BibliographicsNotes . .. .. .. .. .. .. .. .. 182 Chapter8: TolerantLocallyTestableCodes . ......185 8.1 Introduction..................................185 8.2 Preliminaries .................................186 8.3 GeneralObservations . .. .. .. .. .. .. .. .. 188 8.4 TolerantTestersforBinaryCodes . 191 8.4.1 PCPofProximity. .. .. .. .. .. .. .. .191 8.4.2 TheCode ...............................195 8.5 ProductofCodes ...............................200 8.5.1 TolerantTestersforTensorProductsofCodes . .200 8.5.2 RobustTestabilityofProductofCodes . 202 8.6 TolerantTestingofReed-MullerCodes. .205 8.6.1 BivariatePolynomialCodes . 205 8.6.2 GeneralReed-MullerCodes . 207 8.7 BibliographicNotesandOpenQuestions. .209 Chapter9: ConcludingRemarks . 211 9.1 SummaryofContributions . 211 9.2