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Csr2011 june17 09_30_yekhanin
 

Csr2011 june17 09_30_yekhanin

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    Csr2011 june17 09_30_yekhanin Csr2011 june17 09_30_yekhanin Presentation Transcript

    • Locally Decodable Codes
      Sergey Yekhanin
      Microsoft Research
    • Data storage
      • Store data reliably
      • Keep it readily available for users
    • Data storage: Replication
      • Store data reliably
      • Keep it readily available for users
      • Very large overhead
      • Moderate reliability
      • Local recovery:
      Loose one machine, access one
    • Data storage: Erasure coding
      • Store data reliably
      • Keep it readily available for users

      • Low overhead
      • High reliability
      • No local recovery:
      Loose one machine, access k


      k data chunks
      n-k parity chunks
      Need: Erasure codes with local decoding
    • Local decoding: example
      E(X)
      X1
      X2
      X3
      X
      X1
      X2
      X3
      X1X2
      X1X3
      X2X3
      X1X2X3
      • Tolerates 3 erasures
      • After 3 erasures, any information bit can recovered with locality 2
      • After 3 erasures, any parity bit can be recovered with locality 2
    • Local decoding: example
      E(X)
      X1
      X2
      X3
      X
      X1
      X2
      X3
      X1X2
      X1X3
      X2X3
      X1X2X3
      • Tolerates 3 erasures
      • After 3 erasures, any information bit can recovered with locality 2
      • After 3 erasures, any parity bit can be recovered with locality 2
    • Locally Decodable Codes
      Definition : A code is called - locally decodable if for all and all values of the symbol can be recovered from accessing only symbols of , even after an arbitrary 10% of coordinates of are erased.
      k long message
      Decoder reads only r symbols
      Adversarial erasures
      n long codeword
    • Parameters
      Ideally:
      High rate: close to . or
      Strong locality: Very small Constant.
      One cannot minimize and simultaneously. There is a trade-off.
    • Parameters
      Ideally:
      High rate: close to . or
      Strong locality: Very small Constant.
      Potential applications for data transmission / storage.
      Applications in complexity theory / cryptography.
    • Early constructions: Reed Muller codes
      Parameters:
      The code consists of evaluations of all degree polynomials in variables over a finite field
      High rate: No locality at rates above 0.5
      Locality at rate
      Strong locality: for constant
    • State of the art: codes
      High rate: [KSY10]
      Multiplicity codes: Locality at rate
      Strong locality: [Y08, R07, KY09,E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY]
      Matching vector codes: for constant
      for
    • State of the art: lower bounds[KT,KdW,W,W]
      High rate: [KSY10]
      Multiplicity codes: Locality at rate
      Strong locality: [Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11]
      Matching vector codes: for constant
      for
      Locality lower bound:
      Length lower bound:
    • State of the art: constructions
      Matching vector codes
      Reed Muller codes
      Multiplicity codes
    • Plan
      Reed Muller codes
      Multiplicity codes
      Matching vector codes
    • Reed Muller codes
      Parameters:
      Code: Evaluations of degree polynomials over
      Set:
      Polynomial yields a codeword:
      Parameters:
    • Reed Muller codes: local decoding
      Key observation: Restriction of a codeword to an affine line yields an evaluation of a univariate polynomial of degree
      To recover the value at
      Pick a random affine line through
      Do noisy polynomial interpolation.
      • Locally decodable code: Decoder reads random locations.
    • Multiplicity codes
    • Multiplicity codes
      Parameters:
      Code: Evaluations of degree polynomials over
      and their partial derivatives.
      Set:
      Polynomial yields a codeword:
      Parameters:
    • Multiplicity codes: local decoding
      Fact: Derivatives of in two independent directions determine the derivatives in all directions.
      Key observation: Restriction of a codeword to an affine line yields an evaluation of a univariate polynomial of degree
    • Multiplicity codes: local decoding
      To recover the value at
      Pick a line through . Reconstruct
      Pick another line through . Reconstruct
      Polynomials and determine
      • Increasing multiplicity yields higher rate.
      • Increasing the dimension yields smaller query complexity.
    • RM codes vs. Multiplicity codes
    • Matching vector codes
    • Matching vectors
      Definition: Let
      We say that form a matching family if :
      For all
      For all
      Core theorem: A matching vector family of size yields an query code of length
    • MV codes: Encoding
      Let contain a multiplicative subgroup of size
      Given a matching family
      A message:
      Consider a polynomial in the ring:
      • Encoding is the evaluation of over
    • Multiplicity codes: local decoding
      Concept: For a multiplicative line through in direction
      Key observation:evaluation of is a evaluation of a univariate polynomial whose term determines
      To recover
      Pick a random multiplicative line
      Do noisy polynomial interpolation
    • RM codes vs. Multiplicity codes
    • Summary
      Despite progress, the true trade-off between codeword length and locality is still a mystery.
      Are there codes of positive rate with ?
      Are there codes of polynomial length and ?
      A technical question: what is the size of the largest family of subsets of such that
      For all modulo six;
      For all modulo six.