1. Locally Decodable Codes
Sergey Yekhanin
Microsoft Research

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

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

4. 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

5. Local decoding: example
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

6. Local decoding: example
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

7. 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 …

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

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

10. 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

11. 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

12. 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:

13. State of the art: constructions
Matching vector codes
Reed Muller codes
Multiplicity codes

14. Plan
Reed Muller codes
Multiplicity codes
Matching vector codes

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

16. 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.

17. Multiplicity codes

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

19. 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

20. 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.

21. 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

22. Matching vector codes

23. 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

24. 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

25. 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

26. 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

27. 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.