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Locality in Codes and Lifting

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Madhu Sudan

MSR

Joint work with Alan Guo (MIT) and Swastik Kopparty (Rutgers)

Lorentz - Lifted Codes

Lorentz - Lifted Codes

- (Linear) Code .
- block length
- message length
- : Rate of (want as high as possible)
- .

- Basic Algorithmic Tasks
- Encoding: map message in to codeword.
- Testing: Decide if
- Correcting: If , find nearest to .

Lorentz - Lifted Codes

- “Sublinear” time algorithms:
- Algorithms that run in time o(input), o(output).
- Assume random access to input
- Provide random access to output
- Typically probabilistic; allowed to compute output on approximation to input.

- LTCs: Codes that have sublinear time testers.
- Decide if probabilistically.
- Allowed to accept if small.

- LCCs: Codes that have sublinear time correctors.
- If is small, compute , for closest to .

Lorentz - Lifted Codes

- is a -LTC if there exists a tester that
- Makes queries to .
- Accept w.p.
- Reject w.p. at least

- C is a -LCC if exists decoder s.t.
- Given oracle access close to , and
- Decoder makes queries to .
- Decoder usually outputs

- Often: ignore and focus on

Lorentz - Lifted Codes

- Message = multivariate polynomial;
Encoding = evaluations everywhere.

- Locality?
- Restrictions of low-degree
polynomials to lines yield

low-degree (univ.) polys.

- Random lines sample
uniformly (pairwise ind’ly)

- Restrictions of low-degree

Lorentz - Lifted Codes

- Decoding (:
- Problem: Given , , compute .
- Pick random and consider where is a random line .
- Find univ. poly and output

- Testing (:
- Verify

- Parameters:
- ;

Lorentz - Lifted Codes

- Correct more errors (possibly list-decode)
- can correct fraction errors [STV].
- Distance of code
- Decode by projecting to dimensions. “decoding dimension”.
- Locality .
- Lots of work to decode from fraction errors [GKZ,G].
- Open when [Gopalan].

Lorentz - Lifted Codes

- Even slight advantage on test implies correlation with polynomial.[RS, AS]
- Testing dimension ; where ;
- Project to t dimensions and test.
- -LTC.

Lorentz - Lifted Codes

- Why is decoding dimension ?
- Every function on fewer variables is a degree polynomial. So clearly need at least this many dimensions.

- Why is testing dimension ?
- Consider and
- On line ,
- So but has degree on every line!
- In general if then powers of pass through ( … )
- Aside: Using more than testing dimension has not paid dividend with one exception [RazSafra]

Lorentz - Lifted Codes

- Composition of codes yields better LTCs.
- Reduces (to even 3) without too much loss in .
- But till recently,

- LDCs
- Till 2006, multivariate polynomials almost best known.
- 2007+ [Yekhanin, Raghavendra, Efremenko] – great improvements for
- 2010 [KoppartySarafYekhanin] Multiplicity codesget with
- For ; multiv. Polys are still best known.

Lorentz - Lifted Codes

- New Locally Correctible and Testable Codes from “Lifting”.
- for arbitrary
- First “LTCs” to achieve this?
- Only the second “LCCs” with this property
- After Multiplicity codes [KoppartySarafYekhanin]

Lorentz - Lifted Codes

- Alphabet:
- Coordinates:
- Parameter: degree
- Message space:
lines

- Code: Evaluations of message on all of
- And oh …

Lorentz - Lifted Codes

- Functions that look like degree polynomials on every line degree -variate polynomials.
- But this is good news!
- Message space includes all degree d polynomials.
- And has more.
- So rate is higher!
- But does this make a quantitative difference?
- As we will see … YES! Most of the dimension comes from the ``illegitimate’’ functions.

Lorentz - Lifted Codes

- Consider
- extends
- Preferably invariant under affine transformations of .

- Lifted code
- -dim. affine subspaces

- Previous example:

Lorentz - Lifted Codes

- Distance:
- Local Decodability:
- Same decoding algorithm as for RM codes.
- is -LDC implies is -LDC.

- Local Testability?

Lorentz - Lifted Codes

- Local Testability:
- Test: Pick and verify .
- “Single-orbit characterization”: -LTC [KS]
- (Better?) analysis for lifted tests: -LTC [HRS] (extends [BKSSZ,HSS] )

- Musings:
- Analyses not robust (test can’t accept if )
- Still: generalizes almost all known tests … [Main exceptions – [ALMSS,PS,RS,AS] ].
- Key question: what is min s.t.there exists an interpolator s.t.

Lorentz - Lifted Codes

- Distance
- Local Decodability
- Local Testability
- Rate?
- No generic analysis; has to be done on case by case basis.
- Just have to figure out which monomials are in

Lorentz - Lifted Codes

- Will set and let .
- When is ?
- Clearly if ; But that is at most pairs.
- Want more such pairs.
- When is every term of of degree at most ?

Lorentz - Lifted Codes

Notation: if and ( and for all .

Lucas’s Theorem: iff

iff

So given

0

0

0

0

0

msb

lsb

Lorentz - Lifted Codes

Lorentz - Lifted Codes

- Best LCC with O(1) locality.
- ;
- -LCC;

- Alternate codes for BGHMRS construction:

Lorentz - Lifted Codes

- is a Nikodym set if it almost contains a line through every point:
- s.t.

- Similar to Kakeya Set (which contain line in every direction).
- s.t.

- [Dvir], [DKSS]:

Lorentz - Lifted Codes

- Find low-degree poly s.t..
- provided .
- But now Nikodym lines ,
contradicting .

- Conclude .
- Multiplicities, more work, yields .
- But what do we really need from ?
- comes from a large dimensional vector space.
- is low-degree!
- Using from lifted code yields
(provided of small characteristic).

Lorentz - Lifted Codes

- Lifted codes seem to extend “low-degree polynomials” nicely:
- Most locality features remain same.
- Rest are open problems.
- Lead to new codes.

- More generally: Affine-invariant codes worth exploring.
- Can we improve on multiv. poly in polylog locality regime?

Lorentz - Lifted Codes