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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
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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