Sublinear-Time Error-Correction and Error-Detection

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Sublinear-Time Error-Correction and
Error-Detection

• Luca Trevisan
• U.C. Berkeley
• luca_at_eecs.berkeley.edu

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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

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Error-correction
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Error-detection
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Minimum Distance
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Ideally

• Constant information rate
• Linear minimum distance
• Very efficient decoding

Sipser-Spielman linear time deterministic
procedure
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Sub-linear time decoding?

• Must be probabilistic
• Must have some probability of incorrect decoding
• Even so, is it possible?

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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?

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Hadamard Code
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Constant time decoding
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Analysis
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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

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More trade-offs

• For k queries and binary alphabet
• More complicated formulas for bigger alphabet

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Construction without polynomials
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Negative result 1

• Suppose C0,1n -gt 0,1m
• 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 mW(nk/(k-1))
• Katz-Trevisan

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Negative Result 2

• Suppose C0,1n -gt 0,1m
• is linear code with decoding procedure that
  reads only 2 bits of corrupted encoding
• Then there are vectors a1am in 0,1n such that
  for each i1,,n there are W(m) disjoint pairs
  j1,j2 such that
• aj1 xor aj2 ei
• Then it must be mexp(W(n))
• Goldreich-Trevisan, Samorodnitksy

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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

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Bivariate Low Degree Test

• A degree-d bivariate polynomial
• pF x F -gt F is represented as 2F elements of
  Fd (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)

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Analysis

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

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Efficiency of Decoding vs Checking
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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

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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?

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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?

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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?