Succinct representation of codes with applications to testing

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Succinct representation of codes with
applications to testing

• Elena Grigorescu
• Tali Kaufman
• Madhu Sudan

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Outline

• Testing membership in error correcting codes
• Sufficient conditions for testing algebraic codes
• Possible promising perspective rich group of
  symmetries of code
• Our result affine/cyclic invariant, sparse codes
  can be described succinctly by a single, short
  codeword
• Implies locally testability results
• Proof sketch
• Conclusions

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Locally testable codes
0 1 1 0
C

C
C satisfies Code Linear
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Testing linear codes via duality

• BHR Test for linear properties are essentially
• of the form
• Given x, pick
• Accept iff
• Locality of test
• Dual-distance smallest weight of a codeword in
  dual-C

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Sufficient conditions for testing

• Necessary condition for local testing (linear
  codes)
• - small dual distance
• - not sufficient( BHR show random LDPC
  not locally testable)
• Sufficient conditions
• - Possible approach nice symmetries of code
• C is invariant under permutation
  iff

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Symmetries and testing

• Many known testable codes have somewhat large
  symmetry groups
• Eg. Linearity invariance under general linear
  group
• Low degree, Reed-Muller, BCH invariance
  under affine group
• Specific sufficient condition
• KS affine invariance local
  characterization imply testing
• AKKLR Conjecture 2 transitivity small dual
  distance
• Falsified in general GKS
• Modified AKKLR Question What if dual code is
  generated by single low-weight codeword and its
  shifts under some group G (Single-Orbit Property
  under G)
• Are these codes testable (for some group
  G? for all groups G?)

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Single orbit property under affine
invariant/cyclic groups

• Affine group
• Cyclic group
• C has single orbit under cyclic group
• w01001 then B01001, 10100, 01010, 00101,
  10010 is a basis for C
• Formally, C has k-single orbit under G ( included
  in Aut(C) ) if

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Our work

• Study Single-Orbit Property of common codes.
• Def C is sparse if it contains a poly number of
  codewords
• Duals of binary sparse affine invariant codes
  have the single-orbit property under affine group
  
• - under some block-length
  restriction n prime
• - KS08 Single-orbit codes under affine
  group are testable.
• Duals of binary sparse cyclic invariant codes
  have the single-orbit property under cyclic group
  
• - under more block-length
  restrictions n, N-1 primes
• - No testing implications

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Related works

• Sparse, large distance codes are testable
• KL, KS ( tests are coarse, unstructured)
• Affine/linear invariant characterization
  imply testing
• Here sparse
  large distance
• affine invariance
  characterization (explicit tests)
• KL dual-e-BCH codes are testable (unstructured
  tests)
• e-BCH are spanned by shortest codewords
• Here dual-e-BCH are spanned by a single, short
  codeword (explicit basis / tests)

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Toward an explicit description of binary affine
invariant codes

• Affine invariance
• Any function is of the
  form
• The Trace function

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Explicit description of sparse affine families

• Let
• - What aff inv families does f belong to?
• Consider the binary rep of degrees 1, 111,
  1100, 10011
• Then
• In general if degree d occurs then its shadow
  occurs
• Sparsity translates into few monomials
• Affine/Cyclic codes are described by a small set
  of degrees

Shadow(10011) 10011,10010,10001,10000,11,10,1
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Proof ingredients

• Strong number theoretic result of Bourgain
  implies high weight of functions of the form
• few degs gt
  deglt

Degs inside trace
0
?
Weil bounds
Bourgain
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Proof ingredients (contd)

• MacWilliams type counting estimates
• - fourier transform between the functions
  that represent number of codewords for each
  weight in C and in dual- C, respectively
• For sparse codes of length N and of high
  distance obtain

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Proof sketch

• C described by set of degrees D
• Let dual-C Span( aff(w) )
• If C C then there exists
• Let
• Associate C(a) to codew. w
• Does every wtltk codew. belong to a dual of some
  C(a) ?
• New goal exists w that does not belong to the
  dual of any C(a), for all a
• We show

Want exists codew. c with wt lt k s.t.
Span(aff(c))Dual-C
Dual-C
C
C(a)
C
w Dual-C
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Proof Sketch

• C, C(a) sparse, high dist (Bourgain) (assuming
  N-1 and n are primes)
• How many codew of wt k in dual-C?
• How many codew of wt k in dual-C(a) ?
• Total number of degrees a to consider N/n
• Therefore, there exists codew. of wtltk in dual-C
  that whose orbit generates C

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Specifics of the affine case proof

• Here only assume n prime- Bourgain doesnt hold
  for all monomials
• Need codes C(a) to have deg a lt
• Use shadow property
• Show that enough to consider a in the set

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Cyclic codes

• Invariant under
• Punctured affine invariant codes are cyclic
• Cyclic codes are described by generator
  polynomial (or its roots in the field)
• Alternatively described by function families of
  the form
• Degrees can be arbitrary

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Single orbit affine vs cyclic codes

• Affine (length N )
• n prime
• degrees of monomials are shadow closed
• Aut(C)
• single orbit implies testing

• Cyclic (length N-1)
• n, N-1 primes
• degrees of monomials are arbitrary
• Aut(C)N
• not known if single orbit implies testing

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Open Questions

• Do same results hold for non-prime n,
  ?
• Single orbit under what other groups imply
  testing? How large does the Aut group should be
  to imply testing?
• Small weight basis invariance implies testing?
• Examples of families where the tests are not the
  expected ones (I.e. not the ones suggested by
  the description of Aut group)

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Thank you