Sublinear-Time Error-Correction and Error-Detection

1
Sublinear-Time Error-Correction and
Error-Detection

• Luca Trevisan
• U.C. Berkeley
• luca_at_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
is

• Encoding of

0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 1 0 1 0 1 0 1
0 1 0 0 0 1 1 0 0 1 1
0 1 1 0 1 1 0 0 1 1 0
1 0 0 0 0 0 0 1 1 1 1
1 0 1 0 1 0 1 1 0 1 0
1 1 0 0 0 1 1 1 1 0 0
1 1 1 0 1 1 0 1 0 0 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 pFk-gtF
• of degree d (F is a field, F gtgt d)
• Encode message by evaluating p at all Fk 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 d1 points of the line
• by interpolation, find degree-d univariate
  polynomial that agrees with p on the line
• Use interping polynomial to estimate p(x)
• Algorithm reads p in d1 points, each uniformly
  distributed
• Beaver-Feigenbaum Lipton
• Gemmel-Lipton-Rubinfeld-Sudan-Wigderson

19
x(d1)y
x2y
xy
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 pF x F -gt F can
  be represented as 2F univariate degree-d
  polynomials (the rows and the columns)
• 2x2 xy y2 1 mod 5

1 2 0 0 2
3 0 4 0 3
4 2 2 4 3
4 3 4 2 2
3 3 0 4 0
Y21 Y2y3 Y22y4 Y23y4 Y24y3 2x21 2x2x2 2x22x 2x23x 2x24x2
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
  0,1m.
• 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(1o(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?