1. Sublinear-Time Error-Correction and Error-Detection
Luca Trevisan
U.C. Berkeley

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
is…
Encoding of… 1

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 Fk passing 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 2x2 + xy + y2 + 1 mod 5 1 2 3 4
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 1 2 3 4
Y2+1
Y2+y+3
Y2+2y+4
Y2+3y+4
Y2+4y+3
2x2+1
2x2+x+2
2x2+2x
2x2+3x
2x2+4x+2

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 y of 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?