1. of 29 August 4, 2015SIAM AAG: Algebraic Codes and Invariance1 Algebraic Codes and Invariance Madhu Sudan Microsoft Research

2. of 29 Disclaimer Very little Algebraic Geometry in this talk! Very little Algebraic Geometry in this talk! Mainly: Mainly: Coding theorist’s perspective on Algebraic and Algebraic-Geometric Codes Coding theorist’s perspective on Algebraic and Algebraic-Geometric Codes What additional properties it would be nice to have in algebraic-geometry codes. What additional properties it would be nice to have in algebraic-geometry codes. August 4, 2015SIAM AAG: Algebraic Codes and Invariance2 Ex-

3. of 29 Outline of the talk August 4, 2015SIAM AAG: Algebraic Codes and Invariance3

4. of 29 Part 1: Basic Definitions August 4, 2015SIAM AAG: Algebraic Codes and Invariance4

5. of 29 Error-correcting codes August 4, 2015SIAM AAG: Algebraic Codes and Invariance5

6. of 29 Algorithmic tasks August 4, 2015SIAM AAG: Algebraic Codes and Invariance6 Locality

7. of 29 General paradigm: General paradigm: Message space = (Vector) space of functions Message space = (Vector) space of functions Coordinates = Subset of domain of functions Coordinates = Subset of domain of functions Encoding = evaluations of message on domain Encoding = evaluations of message on domain Examples: Examples: Reed-Solomon Codes Reed-Solomon Codes Reed-Muller Codes Reed-Muller Codes Algebraic-Geometric Codes Algebraic-Geometric Codes Others (BCH codes, dual BCH codes …) Others (BCH codes, dual BCH codes …) Algebraic Codes? August 4, 2015SIAM AAG: Algebraic Codes and Invariance7

8. of 29 August 4, 2015SIAM AAG: Algebraic Codes and Invariance8

9. of 29 Essence of combinatorics August 4, 2015SIAM AAG: Algebraic Codes and Invariance9

10. of 29 Consequences August 4, 2015SIAM AAG: Algebraic Codes and Invariance10

11. of 29 August 4, 2015SIAM AAG: Algebraic Codes and Invariance11

12. of 29 Remarkable algorithmics August 4, 2015SIAM AAG: Algebraic Codes and Invariance12

13. of 29 Product property August 4, 2015SIAM AAG: Algebraic Codes and Invariance13

14. of 29 Unique decoding by error-locating pairs August 4, 2015SIAM AAG: Algebraic Codes and Invariance14 [Pellikaan], [Koetter], [Duursma] – 90s

15. of 29 List decoding abstraction August 4, 2015SIAM AAG: Algebraic Codes and Invariance15

16. of 29 August 4, 2015SIAM AAG: Algebraic Codes and Invariance16

17. of 29 Locality in Codes General motivation: General motivation: Does correcting linear fraction of errors require scanning the whole code? Does testing? Does correcting linear fraction of errors require scanning the whole code? Does testing? Deterministically: Yes! Deterministically: Yes! Probabilistically? Not necessarily!! Probabilistically? Not necessarily!! If possible, potentially a very useful concept If possible, potentially a very useful concept Definitely in other mathematical settings Definitely in other mathematical settings PCPs, Small-set expanders, Hardness amplification, Private information retrieval … PCPs, Small-set expanders, Hardness amplification, Private information retrieval … Maybe even in practice Maybe even in practice Aside: Related to LRCs from Judy Walker’s talk. Aside: Related to LRCs from Judy Walker’s talk. Focus here on more errors. Focus here on more errors. August 4, 2015SIAM AAG: Algebraic Codes and Invariance17

18. of 29 Locality of some algebraic codes Locality is a rare phenomenon. Locality is a rare phenomenon. Reed-Solomon codes are not. Reed-Solomon codes are not. Random codes are not. Random codes are not. AG codes are (usually) not. AG codes are (usually) not. Basic examples are algebraic … Basic examples are algebraic … … and a few composition operators preserve it. … and a few composition operators preserve it. Canonical example: Reed-Muller Codes = low- degree polynomials. Canonical example: Reed-Muller Codes = low- degree polynomials. August 4, 2015SIAM AAG: Algebraic Codes and Invariance18

19. of 29 Main Example: Reed-Muller Codes June 16, 2015ISIT: Locality in Coding Theory19

20. of 29 August 4, 2015SIAM AAG: Algebraic Codes and Invariance20

21. of 29 Symmetry in codes August 4, 2015SIAM AAG: Algebraic Codes and Invariance21

22. of 29 Symmetry and Locality August 4, 2015SIAM AAG: Algebraic Codes and Invariance22

23. of 29 Symmetry + Locality - II August 4, 2015SIAM AAG: Algebraic Codes and Invariance23

24. of 29 Aside: Recent Progress in Locality - 1 August 4, 2015SIAM AAG: Algebraic Codes and Invariance24

25. of 29 Aside – 2: Symmetric Ingredients … August 4, 2015SIAM AAG: Algebraic Codes and Invariance25

26. of 29 Part 5: Conclusions August 4, 2015SIAM AAG: Algebraic Codes and Invariance26

27. of 29 Remarkable properties of Algebraic Codes Strikingly strong combinatorially: Strikingly strong combinatorially: Often only proof that extreme choices of parameters are feasible. Often only proof that extreme choices of parameters are feasible. Algorithmically tractable! Algorithmically tractable! The product property! The product property! Surprisingly versatile Surprisingly versatile Broad search space (domain, space of functions) Broad search space (domain, space of functions) August 4, 2015SIAM AAG: Algebraic Codes and Invariance27

28. of 29 Quest for future Construct algebraic geometric codes with rich symmetries. Construct algebraic geometric codes with rich symmetries. In general points on curve have few(er) symmetries. In general points on curve have few(er) symmetries. Can we construct curve carefully? Can we construct curve carefully? Symmetry inherently? Symmetry inherently? Symmetry by design? Symmetry by design? Still work to be done for specific applications. Still work to be done for specific applications. August 4, 2015SIAM AAG: Algebraic Codes and Invariance28

29. of 29 Thank You! August 4, 2015SIAM AAG: Algebraic Codes and Invariance29