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of 30 Locality in Coding Theory Madhu Sudan MSR June 16, 2015ISIT: Locality in Coding Theory1
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of 30 Error-Correcting Codes June 16, 2015ISIT: Locality in Coding Theory2
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of 30 Locality in Algorithms June 16, 2015ISIT: Locality in Coding Theory3
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of 30 LTCs and LCCs: Formally June 16, 2015ISIT: Locality in Coding Theory4
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of 30 Outline of this talk Part 0: Definitions of LTC, LCC Part 1: Elementary construction Part 2: Motivation (historical, current) Part 3: State-of-the-art constructions Part 4 (brief): Towards practicality June 16, 2015ISIT: Locality in Coding Theory5
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of 30 Part 1: Elementary Construction June 16, 2015ISIT: Locality in Coding Theory6
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of 30 Main Example: Reed-Muller Codes June 16, 2015ISIT: Locality in Coding Theory7
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of 30 LDCs and LTCs from Polynomials June 16, 2015ISIT: Locality in Coding Theory8 Analysis non-trivial
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of 30 Part 2: Motivations June 16, 2015ISIT: Locality in Coding Theory9
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of 30 Motivation – 1 (“Practical”) How to encode massive data? – Solution I Encode all data in one big chunk Pro: Pr[failure] = exp(-|big chunk|) Con: Recovery time ~ |big chunk| – Solution II Break data into small pieces; encode separately. Pro: Recovery time ~ |small| Con: Pr[failure] = #pieces X Pr[failure of a piece] – Locality (if possible): Best of both Solutions!! June 16, 2015ISIT: Locality in Coding Theory10
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of 30 Aside: LCCs vs. other Localities Local Reconstruction Codes (LRC): – Recover from few (one? two?) erasures locally. – AND Recover from many errors globally. Regenerating Codes (RgC): – Restricted access pattern for recovery: Partition coordinates and access few symbols per partition. Main Differences: – #errors: LCCs high vs LRC/RgC low – Asymptotic (LCC) vs. Concrete parameters (LRC/RgC) June 16, 2015ISIT: Locality in Coding Theory11
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of 30 Motivation – 2 (“Theoretical”) June 16, 2015ISIT: Locality in Coding Theory12
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of 30 Aside: PCPs (1 of 3) June 16, 2015ISIT: Locality in Coding Theory13
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of 30 PCPs (2 of 3) - Definition June 16, 2015ISIT: Locality in Coding Theory14 W 0 11 V HTHHTH 1.Tosses random coins 2. Determines query locations 3.Reads locations. Accepts/Rejects
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of 30 PCPs (3 of 3): “Polynomial-speak” June 16, 2015ISIT: Locality in Coding Theory15
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of 30 Part 3: Recent Progress on LCCs + LTCs June 16, 2015ISIT: Locality in Coding Theory16
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of 30 Summary of Recent Progress June 16, 2015ISIT: Locality in Coding Theory17
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of 30 Main References Multiplicity codes [KoppartySarafYekhanin’10] See also – Lifted Codes [GuoKoppartySudan’13] – Expander codes [HemenwayOstrovskyWootters’13] Tensor codes [Viderman ‘11] (see also [GKS’13] ) Above + Alon-Luby composition: [KoppartyMeirRon-ZewiSaraf’15] June 16, 2015ISIT: Locality in Coding Theory18 1 3 2
![of 30 Main References Multiplicity codes [KoppartySarafYekhanin’10] See also – Lifted Codes [GuoKoppartySudan’13] – Expander codes [HemenwayOstrovskyWootters’13] Tensor codes [Viderman ‘11] (see also [GKS’13] ) Above + Alon-Luby composition: [KoppartyMeirRon-ZewiSaraf’15] June 16, 2015ISIT: Locality in Coding Theory](http://images.slideplayer.com./19/5775000/slides/slide_18.jpg "of 30 Main References Multiplicity codes [KoppartySarafYekhanin’10] See also – Lifted Codes [GuoKoppartySudan’13] – Expander codes [HemenwayOstrovskyWootters’13] Tensor codes [Viderman ‘11] (see also [GKS’13] ) Above + Alon-Luby composition: [KoppartyMeirRon-ZewiSaraf’15] June 16, 2015ISIT: Locality in Coding Theory")
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of 30 Lifted Codes June 16, 2015ISIT: Locality in Coding Theory19
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of 30 Multiplicity Codes June 16, 2015ISIT: Locality in Coding Theory20
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of 30 Multiplicity Codes - 2 June 16, 2015ISIT: Locality in Coding Theory21
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of 30 State-of-the-art as of 2014 June 16, 2015ISIT: Locality in Coding Theory22
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of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] June 16, 2015ISIT: Locality in Coding Theory23 Short MDS Expander-based Interleaving Message Encoding
![of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] June 16, 2015ISIT: Locality in Coding Theory23 Short MDS Expander-based Interleaving Message Encoding](http://images.slideplayer.com./19/5775000/slides/slide_23.jpg "of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] June 16, 2015ISIT: Locality in Coding Theory23 Short MDS Expander-based Interleaving Message Encoding")
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of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] ISIT: Locality in Coding Theory24 Short MDS Expander-based Interleaving Message Encoding Rate/Distance of final code ~ Rate of MDS [Meir] Locality ~ Locality of Rate 1 code Proof = Picture
![of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] ISIT: Locality in Coding Theory24 Short MDS Expander-based Interleaving Message Encoding Rate/Distance of final code ~ Rate of MDS [Meir] Locality ~ Locality of Rate 1 code Proof = Picture](http://images.slideplayer.com./19/5775000/slides/slide_24.jpg "of 30 Alon-Luby Transformation Key ingredient in [Meir14], [Kopparty et al.’15] ISIT: Locality in Coding Theory24 Short MDS Expander-based Interleaving Message Encoding Rate/Distance of final code ~ Rate of MDS [Meir] Locality ~ Locality of Rate 1 code Proof = Picture")
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of 30 Subpolynomial Locality June 16, 2015ISIT: Locality in Coding Theory25
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of 30 Part 4: Conclusions June 16, 2015ISIT: Locality in Coding Theory26
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of 30 The Locality Advantage June 16, 2015ISIT: Locality in Coding Theory27
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of 30 Locality in Practice? June 16, 2015ISIT: Locality in Coding Theory28
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of 30 Conclusion Locality: (moderately) new model Remarkable effects possible Connect to many other questions in combinatorics/computer science Useful as a data storage mechanism? June 16, 2015ISIT: Locality in Coding Theory29
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of 30 Thank You June 16, 2015ISIT: Locality in Coding Theory30
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