Locally decodable codes allow recovery of individual data symbols even after data loss by accessing only a small number of codeword symbols. Reed-Muller codes provide locality but only up to a rate of 0.5, while multiplicity codes achieve higher rates but have weaker locality guarantees. Matching vector codes can match the best known locality bounds, constructing codes of length n with locality r for constant r, but the optimal tradeoff between rate, length and locality remains an open problem.

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.