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.