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**Locally Decodable Codes**

Sergey Yekhanin Microsoft Research

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Data storage Store data reliably Keep it readily available for users

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**Data storage: Replication**

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

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**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

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**Local decoding: example**

X1X2 X1X3 X2X3 X1X2X3 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

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**Local decoding: example**

X1X2 X1X3 X2X3 X1X2X3 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

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**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 1 … Adversarial erasures n long codeword 1 … 1 …

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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. Potential applications for data transmission / storage. Applications in complexity theory / cryptography.

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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 Locality at rate Strong locality: for constant

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**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

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**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:

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**State of the art: constructions**

Matching vector codes Reed Muller codes Multiplicity codes

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Plan Reed Muller codes Multiplicity codes Matching vector codes

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**Reed Muller codes Parameters:**

Code: Evaluations of degree polynomials over Set: Polynomial yields a codeword:

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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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**Multiplicity codes Parameters:**

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

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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 codes Multiplicity codes Codewords Evaluations of polynomials Higher order evaluations of polynomials Evaluation domain All of the domain All of the domain Decoding Along 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 codes Multiplicity codes Codewords Evaluations of low degree polynomials Evaluations of polynomials with specific monomial degrees Evaluation domain All of the domain A subset of the domain Decoding Along a random affine line Along a random multiplicative line Locally correctable Yes No

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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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