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Locally Decodable Codes Sergey Yekhanin Microsoft Research

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Data storage Store data reliably Store data reliably Keep it readily available for users Keep it readily available for users

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Data storage: Replication Store data reliably Store data reliably Keep it readily available for users Keep it readily available for users Very large overhead Very large overhead Moderate reliability Moderate reliability Local recovery: Local recovery: Loose one machine, access one Loose one machine, access one

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Data storage: Erasure coding Store data reliably Store data reliably Keep it readily available for users Keep it readily available for users Low overhead Low overhead High reliability High reliability No local recovery: No local recovery: Loose one machine, access k Loose one machine, access k … …… k data chunks n-k parity chunks Need: Erasure codes with local decoding

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Local decoding: example X1X1 X1X1 X E(X) 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 X2X2 X2X2 X3X3 X3X3 X1X1 X1X1 X1X2X1X2 X1X2X1X2 X2X2 X2X2 X3X3 X3X3 X1X3X1X3 X1X3X1X3 X2X3X2X3 X2X3X2X3 X1X2X3X1X2X3 X1X2X3X1X2X3

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Local decoding: example X1X1 X1X1 X E(X) 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 X2X2 X2X2 X3X3 X3X3 X1X1 X1X1 X1X2X1X2 X1X2X1X2 X2X2 X2X2 X3X3 X3X3 X1X3X1X3 X1X3X1X3 X2X3X2X3 X2X3X2X3 X1X2X3X1X2X3 X1X2X3X1X2X3

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Locally Decodable Codes Definition : 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 …011 01…01 010…01 k long message n long codeword Adversarial erasures Decoder reads only r symbols

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Parameters Ideally: – High rate: close to. or – Strong locality: Very small Constant. One cannot minimize and simultaneously. There is a trade-off.

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Parameters Ideally: – High rate: close to. or – Strong locality: Very small Constant. Applications in complexity theory / cryptography. Potential applications for data transmission / storage.

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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 High rate: No locality at rates above 0.5 Locality at rate Locality at rate Strong locality: for constant Strong locality: for constant

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State of the art: codes High rate: [KSY10] High rate: [KSY10] Multiplicity codes: L ocality at rate Multiplicity codes: L ocality at rate Strong locality: [Y08, R07, KY09,E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY] Strong locality: [Y08, R07, KY09,E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY] Matching vector codes: for constant Matching vector codes: for constant for for

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State of the art: lower bounds [ KT,KdW,W,W ] High rate: [KSY10] High rate: [KSY10] Multiplicity codes: L ocality at rate Multiplicity codes: L ocality at rate Strong locality: [Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11] Strong locality: [Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11] Matching vector codes: for constant Matching vector codes: for constant for for Length lower bound: Locality lower bound :

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State of the art: constructions Matching vector codesReed Muller codesMultiplicity codes

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Plan Reed Muller codes Reed Muller codes Multiplicity codes Multiplicity codes Matching vector codes Matching vector codes

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Reed Muller codes Parameters: Code: Evaluations of degree polynomials over Set: Polynomial yields a codeword: Parameters:

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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 an affine line through with not too many erasures. – – Do polynomial interpolation. Locally decodable code: Decoder reads random locations.

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Multiplicity codes

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Parameters: Code: Evaluations of degree polynomials over and their partial derivatives. Set: Polynomial yields a codeword: Parameters:

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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

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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.

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RM codes vs. Multiplicity codes Reed Muller codesMultiplicity codes CodewordsEvaluations of polynomials Higher order evaluations of polynomials Evaluation domain All of the domain DecodingAlong a random affine line Along a collection of random affine lines Locally correctable Yes

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Matching vector codes

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Matching vectors Definition: Let We say that form a matching family if : – – For all Core theorem: A matching vector family of size yields an query code of length

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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

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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 multiplicative line – – Do polynomial interpolation

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RM codes vs. Multiplicity codes Reed Muller codesMultiplicity codes CodewordsEvaluations of low degree polynomials Evaluations of polynomials with specific monomial degrees Evaluation domain All of the domainA subset of the domain DecodingAlong a random affine line Along a random multiplicative line Locally correctable YesNo

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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.

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