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An Upper Bound on Locally Recoverable Codes Viveck R. Cadambe (MIT) Arya Mazumdar (University of Minnesota)
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2 Failure Tolerance versus Storage versus Access: Erasure Codes: Classical Trade-off codeword-symbol (storage node)
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3 Failure Tolerance versus Storage versus Access: Erasure Codes: Classical Trade-off codeword-symbol (storage node)
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4 Failure Tolerance versus Storage versus Access: Erasure Codes: Recently studied trade-off codeword-symbol (storage node)
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5 Failure Tolerance versus Storage versus Access*: * Locality important in practice [Huang et. al. 2012, Sathiamoorthy et. al. 2013] * Repair bandwidth is another measure [See a survey by Datta and Oggier 2013] codeword-symbol (storage node) Erasure Codes: Recently studied trade-off
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Trade-off between distance and rate
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Singleton Bound Trade-off between distance and rate Singleton Bound
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Trade-off between distance and rate Singleton Bound
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Trade-off between distance and rate and locality? Singleton Bound
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[Gopalan et. al. 11, Papailiopoulous et. al. 12] Singleton Bound [Gopalan et. al.] Trade-off between distance and rate and locality?
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MRRW Bounds are best known locality- unaware bounds [Gopalan et. al.] MRRW bound Singleton Bound Trade-off between distance and rate and locality? [Gopalan et. al. 11, Papailiopoulous et. al. 12]
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Main Result: A New Upper bound on the price of locality This talk! [Gopalan et. al.] MRRW bound Our Bound
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At least as strong as previously derived bounds. Information theoretic (also applicable for non-linear codes )
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At least as strong as previously derived bounds. Information theoretic (also applicable for non-linear codes ) Analytical insights from Plotkin Bound: Distance-expansion
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At least as strong as previously derived bounds. Information theoretic (also applicable for non-linear codes ) Analytical insights from Plotkin Bound: A bound on the capacity of a particular multicast network for a fixed alphabet (field) size. Because of achievability of [Papailiopoulous et. al. 12] Distance-expansion
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Open Question What is the largest distance achievable by a locally recoverable code, for a fixed alphabet and locality? Our Bound A naïve code A naïve code: Gallager’s LDPC ensemble seems to do better
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Thank you.
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Proof Sketch In the code, t(r+1) nodes that contain tr “q-its of information”, for a certain range of t Remove Locality-induced Redundancy Measure Locality-induced Redundancy
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