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
Most of the results not proved by the speaker
Some of the results not yet proved by anybody

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. Reasons to be interested
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. Hadamard Code

10. “Constant time” decoding

11. Analysis

12. 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

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

14. Construction without polynomials

15. Negative result 1 Suppose C:{0,1}^n -> {0,1}^m
is code with decoding procedure that reads only k bits of corrupted encoding
Pick random x, compute C(x), project C(x) on m^{(k-1)/k} coordinates, prove that it still contains W(n) bits of info. about x.
Then it must be m=W(n^{k/(k-1)})
Katz-Trevisan

16. Negative Result 2 Suppose C:{0,1}^n -> {0,1}^m
is linear code with decoding procedure that reads only 2 bits of corrupted encoding
Then there are vectors a1…am in {0,1}^n such that for each i=1,…,n there are W(m) disjoint pairs j1,j2 such that
aj1 xor aj2 = ei
Then it must be m=exp(W(n))
Goldreich-Trevisan, Samorodnitksy

17. 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

18. Bivariate Low Degree Test
A degree-d bivariate polynomial
p:F x F -> F is represented as 2|F| elements of F^d (the univariate polynomial qa (y) = p(a,y) for each a and the polynomial rb(x) = p(x,b) for each b
Test: pick random a and b, read qa and rb, check that qa(b)=rb(a)

19. Analysis
If |F| is a constant factor bigger than d, then rejection probability is proportional to distance from code
Arora-Safra, ALMSS,
Polishuck-Spielman

20. Efficiency of Decoding vs Checking

21. 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’

22. 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?

23. 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?

24. 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?