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

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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)
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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

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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)
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Weight Distribution of Cr (vs. C)
28
Inner Product with Krawtchouks
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Inner Product with Krawtchouks
Helps! But by how much?
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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)?