Locality in codes and lifting
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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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Locality in Codes and Lifting

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Locality in codes and lifting

Locality in Codes and Lifting

Madhu Sudan

MSR

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

Lorentz - Lifted Codes


Error correcting codes

Error-Correcting 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 .


Locality in algorithms

Locality in Algorithms

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 .


Ltcs and lccs formally

LTCs and LCCs: Formally

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


Example multivariate polynomials

Example: Multivariate Polynomials

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)


Ldcs and ltcs from polynomials

LDCs and LTCs from Polynomials

Lorentz - Lifted Codes

  • Decoding (:

    • Problem: Given , , compute .

    • Pick random and consider where is a random line .

    • Find univ. poly and output

  • Testing (:

    • Verify

  • Parameters:

    • ;


Decoding polynomials

Decoding Polynomials

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].


Testing polynomials

Testing Polynomials

Lorentz - Lifted Codes

  • Even slight advantage on test implies correlation with polynomial.[RS, AS]

  • Testing dimension ; where ;

  • Project to t dimensions and test.

  • -LTC.


Testing vs decoding dimensions

Testing vs. Decoding dimensions

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]


Other ltcs and ldcs

Other LTCs and LDCs

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.


Today

Today

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]


The codes

The codes

Lorentz - Lifted Codes

  • Alphabet:

  • Coordinates:

  • Parameter: degree

  • Message space:

    lines

  • Code: Evaluations of message on all of

  • And oh …


Recall bad news about

Recall: Bad news about

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.


Generalizing lifted codes

Generalizing: Lifted Codes

Lorentz - Lifted Codes

  • Consider

    • extends

    • Preferably invariant under affine transformations of .

  • Lifted code

    • -dim. affine subspaces

  • Previous example:


Properties of lifted codes

Properties of lifted codes

Lorentz - Lifted Codes

  • Distance:

  • Local Decodability:

    • Same decoding algorithm as for RM codes.

    • is -LDC implies is -LDC.

  • Local Testability?


Local testability of lifted codes

Local Testability of lifted codes

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.


Returning to our lifted codes

Returning to (our) lifted codes

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


Rate of bivariate lifted rs codes

Rate of bivariate Lifted RS codes

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 ?


Lucas s theorem rate

Lucas’s theorem & Rate

Lorentz - Lifted Codes

Notation: if and ( and for all .

Lucas’s Theorem: iff

iff

So given


Binary addition etc

Binary addition etc.

0

0

0

0

0

msb

lsb

Lorentz - Lifted Codes


Other lifted codes

Other lifted codes

Lorentz - Lifted Codes

  • Best LCC with O(1) locality.

    • ;

    • -LCC;

  • Alternate codes for BGHMRS construction:


Nikodym sets

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]:


Proof polynomial method

Proof (“Polynomial Method”)

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).


Conclusions

Conclusions

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?


Thank you

Thank You

Lorentz - Lifted Codes


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