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Locally correctable codes from lifting Alan Guo MIT CSAIL Joint work with Swastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)

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Presentation on theme: "Locally correctable codes from lifting Alan Guo MIT CSAIL Joint work with Swastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)"— Presentation transcript:

1 Locally correctable codes from lifting Alan Guo MIT CSAIL Joint work with Swastik Kopparty (Rutgers) and Madhu Sudan (Microsoft Research)

2 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework of lifting codes – New lower bounds for Nikodym sets

3 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework of lifting codes – New lower bounds for Nikodym sets

4 Error correcting codes

5 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

6 Locality

7 Bivariate polynomial codes

8 Local correctability

9 A brief history of LCCs

10 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

11 Our contributions

12 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

13 Main idea

14

15

16 Dimension of lifted RS code

17

18

19 Reed-Muller Lifted Reed-Solomon

20 Dimension of lifted RS code Reed-Muller Lifted Reed-Solomon

21 Dimension of lifted RS code Reed-Muller Lifted Reed-Solomon

22 Dimension of lifted RS code Reed-Muller Lifted Reed-Solomon

23 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

24 General results

25 Talk outline Error correcting codes Locally correctable codes Our contributions – New high rate LCCs – General framework for lifting codes – New lower bounds for Nikodym sets

26 Application to Nikodym sets

27

28

29 Summary Lifting – Natural operation – Build longer codes from short ones – Preserve distance – Gain locality – Can get high rate Applications outside of coding theory – Improve polynomial method (e.g. Nikodym sets)

30 Thank you!

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