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# Locality in Codes and Lifting - PowerPoint PPT Presentation

Locality in Codes and Lifting. Madhu Sudan MSR. Joint work with Alan Guo (MIT) and Swastik Kopparty (Rutgers). Error-Correcting Codes. (Linear) Code . block length message length : Rate of (want as high as possible) . Basic Algorithmic Tasks

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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)

• .

• 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).

• 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)

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:

Nikodym Sets

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 ,

• 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.