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An Upper Bound on Locally Recoverable Codes

An Upper Bound on Locally Recoverable Codes. Viveck R. Cadambe (MIT) Arya Mazumdar (University of Minnesota). Erasure Codes: Classical Trade-off. Failure Tolerance versus Storage versus Access: . codeword -symbol (storage node). Erasure Codes: Classical Trade-off.

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An Upper Bound on Locally Recoverable Codes

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  1. An Upper Bound on Locally Recoverable Codes Viveck R. Cadambe (MIT) AryaMazumdar (University of Minnesota)

  2. Erasure Codes: Classical Trade-off Failure Tolerance versus Storage versus Access: codeword-symbol (storage node)

  3. Erasure Codes: Classical Trade-off Failure Tolerance versus Storage versus Access: codeword-symbol (storage node)

  4. Erasure Codes: Recently studied trade-off Failure Tolerance versus Storage versus Access: codeword-symbol (storage node)

  5. Erasure Codes: Recently studied trade-off Failure Tolerance versus Storage versus Access*: codeword-symbol (storage node) * Locality important in practice [Huang et. al. 2012, Sathiamoorthy et. al. 2013] * Repair bandwidth is another measure [See a survey by Datta and Oggier2013]

  6. Trade-off between distance and rate

  7. Trade-off between distance and rate Singleton Bound Singleton Bound

  8. Trade-off between distance and rate Singleton Bound Singleton Bound

  9. Trade-off between distance and rate and locality? Singleton Bound Singleton Bound

  10. Trade-off between distance and rate and locality? [Gopalan et. al.] Singleton Bound [Gopalan et. al. 11, Papailiopoulouset. al. 12] Singleton Bound

  11. Trade-off between distance and rate and locality? MRRW bound [Gopalan et. al.] Singleton Bound [Gopalan et. al. 11, Papailiopoulouset. al. 12] MRRW Bounds are best known locality-unaware bounds

  12. Main Result: A New Upper bound on the price of locality MRRW bound [Gopalan et. al.] This talk! Our Bound

  13. At least as strong as previously derived bounds. • Information theoretic (also applicable for non-linear codes )

  14. At least as strong as previously derived bounds. • Information theoretic (also applicable for non-linear codes ) • Analytical insights from Plotkin Bound: Distance-expansion

  15. 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

  16. 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

  17. Thank you.

  18. Proof Sketch Measure Locality-induced Redundancy In the code, t(r+1) nodes that contain tr “q-its of information”, for a certain range of t Remove Locality-induced Redundancy

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