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Explore the concept of locality in coding theory, focusing on efficiency and structure. Learn about classical and modern developments, including local decoding and reconstruction codes. Discover how locality impacts data protection and storage strategies.
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Locality in Coding Theory MadhuSudan Harvard University Wola'16: Locality in Coding Theory
Locality LOCALity • Local “Influenced by small neighborhood” • Two views in literature: • Focus on “small” • Leads to greater efficiency in algorithms, but not necessarily structure. • Focus on “neighborhood” • Suitable in distributed computing. • More structure in algorithms • Input to algorithm = architecture for solution • Coding Theory: Focus on “small” Wola'16: Locality in Coding Theory
Locality in Coding Theory • Historical accident? • Classical concepts: • Local decodability • Local list-decodability • Local testability • Striking recent developments: • Modern Concepts: • Local Reconstruction Codes • Some constructions Wola'16: Locality in Coding Theory
(Classical) Challenge of Coding Theory • Protect data from (inevitable) error • Question: Does ability to protect go with size • Shannon’s surprising answer (surprise in 1948): YES • Cost = ? Recovery time with size • Current options to store massive data: • Break data into small chunks. • Good recovery time but lose twice in reliability: (1) Reliability per chunk; (2) Union bound on chunks. • Keep data in one big chunk: • Good reliability, but poor recovery time. • Locality (if achievable): Best of both worlds. • Recovery time = sublinear? Wola'16: Locality in Coding Theory
Basic Notions: • Code: ; • Testing: Given ; s.t. ? • Decoding: Given find s.t. • Locality: Do above in sublinear time! • Local testing: • Make few queries to ; • Reject with prob. • Locally Decodable (correctible) Code: • Given make few queries to ; output • Local Reconstruction Codes: • LDCs for ; (even simpler: erasures only!) Wola'16: Locality in Coding Theory
Progress Report: LTCs • Motivations: • Entirely mathematical? • PCPs … XOR lemma, extremal graphs, … • Initial belief:Locality as ; • Disproved many times: • [GS,BGHSV,BS,Dinur,Meir]: with • Semi-recent belief: Locality as • Disproved fewer times: • [Viderman’11]: and , . • [Kopparty,Meir,Ron-Zewi,Saraf’16]: achievable for “free” • Practical use? Wola'16: Locality in Coding Theory
Progress Report: LDCs • Motivations: • Mathematical … + (in hindsight) Practical? • Initial belief:Locality as ; • Disproved (2006-9) [Yekhanin, Raghavendra, Efremenko]: • Semi-recent belief: Locality as • Disproved “many times” (2010-2016): • [Kopparty,Meir,Ron-Zewi,Saraf’16]: achievable for “free” • Practical Use?! Wola'16: Locality in Coding Theory
Local Reconstruction Codes • Practically motivated notion: • Fix small corruption quickly (locally). • Remain resilient to many errors. • -LRC [Gopalan-Huang-Simitci-Yekhanin]: • Code of distance s.t.single symbol erasure can be recovered -locally. • Symbol of message? (Weak Defn.) • Symbol of codeword? (Strong Defn.) Wola'16: Locality in Coding Theory
An Easy Construction Standard ECC (systematic) Reed-Solomon Satisfies weak defn. Can save if it is of all Wola'16: Locality in Coding Theory
A limiting result • Recall Singleton Bound: • Project Code to first coordinates. • Pigeonhole Principle s.t. • Simple -LRC weaker by • Thm [GHSY]: This is necessary (up to ) • Find locations of rank greedily. • Project to these coordinates+ PHP. Wola'16: Locality in Coding Theory
A “state-of-art” construction • [Tamo-Barg]: (Strong Defn.) • Code = Subcode of Reed-Solomon Code: • Work with where divides ; • () • Local groups:; points • ; deg. ! Wola'16: Locality in Coding Theory
Conclusions • Locality in coding theory • Diverse collection of notions • New problems and solutions (and techniques). • Diverse applications. • Many open questions! Wola'16: Locality in Coding Theory