Presentation on theme: "Locally Decodable Codes"— Presentation transcript:
1Locally Decodable Codes Sergey YekhaninMicrosoft Research
2Data storageStore data reliablyKeep it readily available for users
3Data storage: Replication Store data reliablyKeep it readily available for usersVery large overheadModerate reliabilityLocal recovery:Loose one machine, access one
4Data storage: Erasure coding Store data reliablyKeep it readily available for users…Low overheadHigh reliabilityNo local recovery:Loose one machine, access k……k data chunksn-k parity chunksNeed: Erasure codes with local decoding
5Local decoding: example X1X2X1X3X2X3X1X2X3Tolerates 3 erasuresAfter 3 erasures, any information bit can recovered with locality 2After 3 erasures, any parity bit can be recovered with locality 2
6Local decoding: example X1X2X1X3X2X3X1X2X3Tolerates 3 erasuresAfter 3 erasures, any information bit can recovered with locality 2After 3 erasures, any parity bit can be recovered with locality 2
7Locally 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 messageDecoder reads only r symbols1…Adversarial erasuresn long codeword1…1…
8ParametersIdeally:High rate: close to orStrong locality: Very small Constant.One cannot minimize and simultaneously. There is a trade-off.
9Parameters Ideally: High rate: close to . or Strong locality: Very small Constant.Potential applications for data transmission / storage.Applications in complexity theory / cryptography.
10Early constructions: Reed Muller codes Parameters:The code consists of evaluations of all degree polynomials in variables over a finite fieldHigh rate: No locality at rates above 0.5Locality at rateStrong locality: for constant
11State of the art: codes High rate: [KSY10] Multiplicity codes: Locality at rateStrong locality: [Y08, R07, KY09,E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY]Matching vector codes: for constantfor
12State of the art: lower bounds [KT,KdW,W,W] High rate: [KSY10]Multiplicity codes: Locality at rateStrong locality: [Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11]Matching vector codes: for constantforLocality lower bound:Length lower bound:
13State of the art: constructions Matching vector codesReed Muller codesMultiplicity codes
15Reed Muller codes Parameters: Code: Evaluations of degree polynomials overSet:Polynomial yields a codeword:
16Reed Muller codes: local decoding Key observation: Restriction of a codeword to an affine line yields an evaluation of a univariate polynomial of degreeTo recover the value atPick an affine line through with not too many erasures.Do polynomial interpolation.Locally decodable code: Decoder reads random locations.
18Multiplicity codes Parameters: Code: Evaluations of degree polynomials overand their partial derivatives.Set:Polynomial yields a codeword:
19Multiplicity 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
20Multiplicity codes: local decoding To recover the value atPick a line through ReconstructPick another line through ReconstructPolynomials and determineIncreasing multiplicity yields higher rate.Increasing the dimension yields smaller query complexity.
21RM codes vs. Multiplicity codes Reed Muller codesMultiplicity codesCodewordsEvaluations of polynomialsHigher order evaluations of polynomialsEvaluation domainAll of the domainAll of the domainDecodingAlong a random affine lineAlong a collection of random affine linesLocally correctableYes
23Matching vectors Definition: Let We say that form a matching family if :For allCore theorem: A matching vector family of size yields an query code of length
24MV codes: Encoding Let contain a multiplicative subgroup of size Given a matching familyA message:Consider a polynomial in the ring:Encoding is the evaluation of over
25Multiplicity codes: local decoding Concept: For a multiplicative line through in directionKey observation: evaluation of is a evaluation of a univariate polynomial whose term determinesTo recoverPick a multiplicative lineDo polynomial interpolation
26RM codes vs. Multiplicity codes Reed Muller codesMultiplicity codesCodewordsEvaluations of low degree polynomialsEvaluations of polynomials with specific monomial degreesEvaluation domainAll of the domainA subset of the domainDecodingAlong a random affine lineAlong a random multiplicative lineLocally correctableYesNo
27SummaryDespite 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 thatFor all modulo six;For all modulo six.