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Simple PCPs

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Joint work with Tali Kaufman (IAS & MIT). Local (Sublinear-time) Algorithmics. Data getting ever-larger ... Exception: [Meir '08] not algebraic. Questions: ... – PowerPoint PPT presentation

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Title: Simple PCPs


1
Local Testability and Decodability of Sparse
Linear Codes
Madhu Sudan MIT
Joint work with Tali Kaufman (IAS MIT).
2
Local (Sublinear-time) Algorithmics
  • Data getting ever-larger
  • Need algorithms that can infer global
    properties from local observations
  • Led to
  • Property testing, Sublinear-time algorithms
  • Common themes
  • Oracle-access to input, implicit output.
  • Answers of the form input close to having
    property

3
Error-Correcting Codes
  • Code
  • Distance
  • between sequences
  • of code
  • Algorithmic Problems
  • Encode
  • Detect Errors
  • Decode

4
Local Algorithmics in Coding
  • Encoding Can not be performed locally
  • Single bit change in input should alter constant
    fraction of output!
  • Testing, Decoding, Error-correcting can be
    performed locally. Furthermore
  • They are very natural problems.
  • Have many applications in theory (PCP, PIR,
    Hardness amplification).
  • Lots of interesting effects are achievable.

5
Local Algorithmic Problems
  • Common framework
  • Local Testing
  • Local Self-Correction
  • Local Decoding

6
Example Hadamard Codes
  • Encoding
  • Test
  • Correction
  • Decoding

7
Brief History
  • Local Decoding/Self-Correcting
  • Beaver-Feigenbaum, Lipton, Blum-Luby-Rubinfel
    d instances of Local Decodability.
  • Katz-Trevisan first definition.
  • Locally Testable Codes
  • Blum-Luby-Rubinfeld, Babai-Fortnow-Lund
    first instances.
  • Arora, Rubinfeld-Sudan, Spielman,
    Goldreich-Sudan definitions.

8
Constructions of Locally X-able Codes
  • Basic codes Algebraic in nature.
  • Analysis
  • Decoding typically simple, uses algebra.
  • Testing more complex.
  • Better codes Careful compositions of basic
    codes.
  • Exception Meir 08 not algebraic.
  • Questions
  • Do we need all this algebra/careful
    constructions?
  • Can we derive local algorithms from classical
    parameters?
  • Can randomly chosen codes have local algorithms?

9
Our Results
  • Theorem (Informal) Every sparse, linear code
    of large distance is locally testable,
    correctible.
  • Linear?
  • Sparse?
  • Large Distance?

10
Our Results (contd.)
  • Linear?
  • Sparse?
  • Large Distance?
  • Balanced?

11
Corollaries
  • Reproduce old results Hadamard, dual-BCH
  • New codes
  • Random sparse linear codes (decodable under any
    linear encoding).
  • dual-BCH variants
  • Nice closure properties (Subcodes, Addition of
    new coordinates, removal of few coordinates)

12
Previously
  • Kaufman-Litsyn Similar result techniques.
    Main differences
  • Required
  • Worked only for balanced codes.
  • Only proved local testability no correctibility

13
Proof Techniques
  • Modifying (simplifying? extending?) the proofs of
    Kaufman Litsyn 05 (some ideas go back to Kiwi
    95).
  • Buzzwords Duality, MacWilliams Identities,
    Krawtchouk Polynomials, Johnson bounds.

14
Duality Testing
  • Dual of a Code
  • Canonical (only) test for membership in C
  • Canonical self-corrector

15
Questions

16
Path to answers
  • Need weight distribution of some codes
  • Testing Correcting
  • Testing Kiwi, KL
  • Correcting New

17
Dual Weight Distribution?
  • MacWilliams Identities Can compute weight
    distribution of dual from weight distribution of
    primal exactly!
  • Dont have primal distribution exactly Can
    coarse information suffice?
  • Kiwi - Manages to compute primal info. exactly.
  • Kaufman-Litsyn Find out a lot about primal
    distribution.
  • Our hope Less precise info. sufficient.

18
MacWilliams Identities Precise Form
  • Krawtchouk Polynomials
  • Dual Weight Distribution
  • Double summation! Many negative terms.
    Cancellations?

19
Primal Weight Distribution (Balanced)
20
Krawtchouk Polynomial (k odd)
21
Krawtchouk Polynomial (k odd)
22
Low-weight codewords in dual
  • Can conclude constant weight codewords exist.
  • Very tight bound
  • Leads to self-corrector

23
Analysis of self-corrector
  • Need to understand
  • New Code
  • Claim
  • But

24
Analysis of self-corrector (contd.)
  • Plugging in bounds
  • Similar calculations with yield
  • Conclude Self-corrector computes
    correctly w.p. from
    -corrupted received word.

25
Analysis of Tester (balanced case)
  • Need to analyze
  • Specifically, want
  • Easy fact (from MacWilliams Identities)
  • Suffices to analyze second term. But what does
    the weight distribution of look like?
    and how does interact with this?

26
Weight Distribution of Cr (vs. C)
27
Weight Distribution of Cr (vs. C)
28
Inner Product with Krawtchouks
29
Inner Product with Krawtchouks
Helps! But by how much?
30
More Bounds
  • Some weak Krawtchouk bounds
  • Bound 2. not sufficient to bound the hurt but
    can combine with Johnson bound
  • Johnson Bound

(the helpful part)
(For i in our range. Useful to limit the hurt)
31
Putting all the bounds together
  • Can conclude
  • Implies test rejects -corrupted codeword with
    probability

32
Unbalanced codes?
  • Many things breakdown
  • E.g.,
  • Our approach
  • Step 1
  • Step 2

33
Weakly balanced codes
  • Can now prove
  • But cant get a precise bound on
  • Instead, we bound
    directly
  • Show that contribution of any word to both terms
    is roughly the same (Uses some properties of
    .)
  • Show that contribution of the coset leader drops
    by -factor.

34
Reducing general codes to w.b. codes
  • Write where is
    weakly-balanced.
  • Test if such that
  • Yields tester for all binary, linear, sparse,
    high-distance codes.

35
Conclusions/Questions
  • Simpler proof for random codes by Shachar Lovett,
    Or Meir.
  • Self-correct imbalanced codes?
  • Are random sparse codes locally list-decodable?
  • Is this just a logarithmic saving in locality?
  • Are there other ways to pick broad classes of
    testable codes (at random)?
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