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 ID: 276154
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Slide1
Locally Decodable Codes
Sergey Yekhanin
Microsoft ResearchSlide2
Data storage
Store data reliably
Keep it readily available for usersSlide3
Data storage: Replication
Store data reliably
Keep it readily available for users
Very large overhead
Moderate reliability
Local recovery
:
Loose one machine, access oneSlide4
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 decodingSlide5
Local decoding: example
X
1
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
X
2
X
3
X
1
X
1
X
2
X
2X3
X1X3
X2X3X1
X2X3Slide6
Local decoding: example
X
1
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
X
2
X
3
X
1
X
1
X
2
X
2X3X1X3
X2X3
X1X2X3Slide7
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.
0
0
1
0
1
…
0
1
1
0
1
…
0
1
0
1
0
…
0
1
k
long
message
n
long
codeword
Adversarial
erasures
Decoder reads only
r
symbolsSlide8
Parameters
Ideally:
High rate: close to . or Strong locality: Very small Constant.
One cannot minimize and simultaneously. There is a trade-off.Slide9
Parameters
Ideally:
High rate: close to . or Strong locality: Very small Constant.
Applications in complexity theory / cryptography.
Potential applications for data transmission / storage.Slide10
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 Slide11
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
Slide12
State of the art: lower bounds
[KT,KdW,W,W]
High rate: [KSY10]
Multiplicity codes:
L
ocality at rate
Strong locality
:
[Y08, R07, E09, DGY10, BET10a, IS10, CFL+10,BET10b,SY11]
Matching vector codes: for constant
for
Length lower bound:
Locality lower bound
:Slide13
State of the art: constructions
Matching vector codes
Reed Muller codes
Multiplicity codesSlide14
Plan
Reed Muller codes
Multiplicity codesMatching vector codesSlide15
Reed Muller codes
Parameters:
Code: Evaluations of degree polynomials over Set: Polynomial yields a codeword:Parameters:Slide16
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.Slide17
Multiplicity codesSlide18
Multiplicity codes
Parameters:
Code: Evaluations of degree polynomials over and their partial derivatives.Set:Polynomial yields a codeword:
Parameters:
Slide19
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 Slide20
Multiplicity codes: local decoding
To recover the value at
Pick a line through . ReconstructPick another line through . ReconstructPolynomials and determine
Increasing multiplicity yields higher rate.
Increasing the dimension yields smaller query complexity.
Slide21
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
YesSlide22
Matching vector codesSlide23
Matching vectors
Definition
: Let We say that form a matching family if :For allFor all
Core theorem
: A matching vector family of size yields an query code of length Slide24
MV 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
Slide25
Multiplicity codes: local decoding
Concept
: For a multiplicative line through in directionKey observation
:
evaluation of is a evaluation of a univariate polynomial whose term determines
To recover
Pick a
multiplicative
line
Do
polynomial
interpolation
Slide26
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
NoSlide27
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.