Sublinear-Time Error-Correction and Error-Detection

1. Sublinear-Time Error-Correction and Error-Detection Luca Trevisan U.C. Berkeley luca@eecs.berkeley.edu

2. Contents • Survey of results on error-correcting codes with sub-linear time checking and decoding procedures • Results originated in complexity theory

3. Error-correction

4. Error-detection

5. Minimum Distance

6. Ideally • Constant information rate • Linear minimum distance • Very efficient decoding Sipser-Spielman: linear time deterministic procedure

7. Sub-linear time decoding? • Must be probabilistic • Must have some probability of incorrect decoding • Even so, is it possible?

8. Motivations & Context • Sub-linear time decoding useful for worst-case to average-case reductions, and in information-theoretic Private Information Retrieval • Sub-linear time checking arises in PCP • Useful in practice?

9. Error-correction

10. Hadamard Code

11. Example Encoding of… is…

12. “Constant time” decoding

13. Analysis

14. A Lower Bound • If: the code is linear, the alphabet is small, and the decoding procedure uses two queries • Then exponential encoding length is necessary Goldreich-Trevisan, Samorodnitsky

15. More trade-offs • For k queries and binary alphabet: • More complicated formulas for bigger alphabet

16. Construction without polynomials

17. Construction with polynomials • View message as polynomial p:Fk->F of degree d (F is a field, |F| >> d) • Encode message by evaluating p at all |F|k points • To encode n-bits message, can have |F| polynomial in n, and d,k around (log n)O(1)

18. To reconstruct p(x) • Pick a random line in Fkpassing through x; • evaluate p on d+1 points of the line; • by interpolation, find degree-d univariate polynomial that agrees with p on the line • Use interp’ing polynomial to estimate p(x) • Algorithm reads p in d+1 points, each uniformly distributed Beaver-Feigenbaum; Lipton; Gemmel-Lipton-Rubinfeld-Sudan-Wigderson

19. x+(d+1)y x+2y x+y x

20. Error-detection

21. Checking polynomial codes • Consider encoding with multivariate low-degree polynomials • Given p, pick random z, do the decoding for p(z), compare with actual value of p(z) • “Simple” case of low-degree test. • Rejection prob. proportional to distance from code. Rubinfeld-Sudan

22. Bivariate Code • A degree-d bivariate polynomial p:F x F -> F can be represented as 2|F| univariate degree-d polynomials (the “rows” and the columns”) 2x2 + xy + y2 + 1 mod 5

23. Bivariate Low-Degree Test • Pick a random row and a random column. Chek that they agree on intersection • If |F| is a constant factor bigger than d, then rejection probability is proportional to distance from code Arora-Safra, ALMSS, Polishuck-Spielman

24. Efficiency of Decoding vs Checking

25. Tensor Product Codes • Suppose we have a linear code C with codewords in {0,1}^m. • Define new code C’ with codewords in {0,1}^(mxm); • a “matrix” is a codeword of C’ if each row and each column is codeword for C • If C has lots of codeword and large minimum distance, same true for C’

26. Generalization of the Bivariate Low Degree Test • Suppose C has K codewords • Define code C’’ over alphabet [K], with codewords of length 2m • C’’ has as many codewords as C’ • For each codeword yof C’, corresponding codeword in C’’ contains value of each row and each column of y • Test: pick a random “row” and a random “column”, check intersection agrees • Analysis?

27. Negative Results? • No known lower bound for locally checkable codes • Possible to get encoding length n^(1+o(1)) and checking with O(1) queries and {0,1} alphabet? • Possible to get encoding length O(n) with O(1) queries and small alphabet?

28. Applications? • Better locally decodable codes have applications to PIR • General/simple analysis of checkable proofs could have application to PCP (linear-length PCP, simple proof of the PCP theorem) • Applications to the practice of fault-tolerant data storage/transmission?