1. Locally Decodable Codes Sergey Yekhanin Microsoft Research

4. Keep it readily available for users

5. Very large overhead

6. Moderate reliability

7. Local recovery: Loose one machine, access one

10. High reliability

11. No local recovery: Loose one machine, access k … … k data chunks n-k parity chunks Need: Erasure codes with local decoding

13. After 3 erasures, any information bit can recovered with locality 2

15. After 3 erasures, any information bit can recovered with locality 2

17. Parameters Ideally: High rate: close to . or Strong locality: Very small Constant. One cannot minimize and simultaneously. There is a trade-off.

18. Parameters Ideally: High rate: close to . or Strong locality: Very small Constant. Potential applications for data transmission / storage. Applications in complexity theory / cryptography.

19. Early constructions: Reed Muller codes Parameters: The code consists of evaluations of all degree polynomials in variables over a finite field High rate: No locality at rates above 0.5 Locality at rate Strong locality: for constant

20. State of the art: codes High rate: [KSY10] Multiplicity codes: Locality at rate Strong locality: [Y08, R07, KY09,E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY] Matching vector codes: for constant for

21. State of the art: lower bounds[KT,KdW,W,W] High rate: [KSY10] Multiplicity codes: Locality at rate Strong locality: [Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11] Matching vector codes: for constant for Locality lower bound: Length lower bound:

22. State of the art: constructions Matching vector codes Reed Muller codes Multiplicity codes

23. Plan Reed Muller codes Multiplicity codes Matching vector codes

24. Reed Muller codes Parameters: Code: Evaluations of degree polynomials over Set: Polynomial yields a codeword: Parameters:

26. Multiplicity codes Parameters: Code: Evaluations of degree polynomials over and their partial derivatives. Set: Polynomial yields a codeword: Parameters:

27. Multiplicity codes: local decoding Fact: Derivatives of in two independent directions determine the derivatives in all directions. Key observation: Restriction of a codeword to an affine line yields an evaluation of a univariate polynomial of degree

30. Matching vector codes

31. Matching vectors Definition: Let We say that form a matching family if : For all For all Core theorem: A matching vector family of size yields an query code of length

33. RM codes vs. Multiplicity codes

34. Summary Despite progress, the true trade-off between codeword length and locality is still a mystery. Are there codes of positive rate with ? Are there codes of polynomial length and ? A technical question: what is the size of the largest family of subsets of such that For all modulo six; For all modulo six.