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

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

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

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

5. 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

6. Trade-off between distance and rate

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

8. Trade-off between distance and rate Singleton Bound

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

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

11. 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]

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