1. List decoding Reed-Muller codes up to minimal distance: Structure and pseudo- randomness in coding theory Abhishek Bhowmick (UT Austin) Shachar Lovett (UC San Diego)

2. What this talk is about? Technically: new understanding of a basic and important family of codes Conceptually: structure and pseudo-randomness play important roles in many computational domains. This talk shows this phenomena applied to coding theory

3. Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

4. Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

5. Decoding from errors The basic problem of coding theory: recovering from errors Goal: recover correct codeword from a noisy received word This work: worst-case errors

6. Unique decoding Codeword Received word

7. Unique decoding Unique decoding: find the closest codeword Basic limitation: minimal distance of the code If a received word is “in between” two codewords, we cannot distinguish which is the correct codeword Limits error to <½ the minimal distance

8. Unique decoding Codeword Received word

9. List decoding List decoding: find few closest codewords [Elias ‘57] Circumvents the ½ minimal distance problem In general, can recover from errors up to Johnson bound ½ minimal distance < Johnson bound < minimal distance For special codes (hmmm…) can do better

10. List decoding Codeword Received word

11. Polynomial codes Most codes are based on polynomials In this talk, focus on the most basic families Reed-Solomon: univariate polynomials Reed-Muller: multivariate polynomials Despite (or because) being basic, they are widely applied; however, they are far from fully understood

12. Why polynomial codes? Polynomial codes are “special” Do they behave better than “worst-case” analysis? Concretely: are they list decodable beyond the Johnson bound? Previous works: yes This work: YES

13. Reed-Muller codes

14. Minimal distance of Reed-Muller codes

16. Hadamard codes Hadamard codes correspond to d=1 (linear functions) Minimal distance=1/|F| List decodable up to error 1/|F| [Goldreich-Levin’89, Goldreich-Rubinfeld-Sudan’00] Proof: Fourier analysis

17. Large fields

18. Small fields Breakthrough in 2008: Over F 2, RM codes are list decodable up to minimal distance (combinatorially & algorithmically) [Gopalan-Klivans-Zuckerman’08] Proof doesn’t extend to larger fields: uses special properties of Johnson bound over binary fields GKZ conjecture: all RM codes are list decodable up to minimal distance (they are all “special”)

19. GKZ conjecture

20. Main result (this work)

21. Extension to large fields (in progress)

22. Proof idea

23. Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

24. The list decoding problem, revisited

25. Regularity for list decoding Lemma: for any code, any received word can be replaced by a “low complexity” received word, which is indistinguishable from the code perspective Similar to the Frieze-Kannan weak regularity Viewpoint: codewords are “tests”

26. Regularity for list decoding

27. Codeword Original received word Low complexity received word

28. Proof of regularity lemma

29. Upshot

30. Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

31. A very special case (which will turn out to be not so special)

32. Rethinking minimal distance

33. A structural lemma

34. Proof of structural lemma

35. Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

36. Pseudo-random polynomials

37. Pseudo-random polynomials: examples

38. A dichotomy theorem

39. Decompositions of polynomials The dichotomy theorem can be applied iteratively, to decompose any low-degree polynomial as a function of a few polynomials which are pseudo-random To a large extent, pseudo-random polynomials behave as “independent variables” Made precise in higher-order Fourier analysis

40. Going back to the very special case

41. Overview Coding theory 101 Regularity in coding theory Structural properties of polynomials Pseudo-randomness for polynomials Summary

42. Result Reed-Muller codes are special: can be list decoded up to minimal distance (for constant degrees, fields) Proof relies on three ingredients: 1.Regularity for codes 2.Structural property of RM codes (polynomials) 3.Pseudo-randomness for RM codes (polynomials)

43. Follow up work We extend the current result to the case of large fields This requires a few new ingredients: 1.Optimizing the arguments, to get a polynomial dependency on the field size 2.Extending higher-order Fourier to large fields, with bounds independent of field size

44. Take home message Notions of structure and pseudo-randomness are very powerful; dichotomy theorems make them universal This work: coding theory, applied to RM codes Other applications: math - graph theory, number theory, ergodic theory, discrete geometry; CS - property testing, complexity, algorithms Question: do our techniques generalize to other codes? Other domains?

45. Thank you!