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

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Presentation transcript:

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

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

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

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

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

Trade-off between distance and rate

Singleton Bound Trade-off between distance and rate Singleton Bound

Trade-off between distance and rate Singleton Bound

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

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

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]

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

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

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

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

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

Thank you.

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