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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). 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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  1. Locality in Codes and Lifting Madhu Sudan MSR Joint work with Alan Guo (MIT) and Swastik Kopparty (Rutgers) Lorentz - Lifted Codes

  2. 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 .

  3. 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 .

  4. 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

  5. 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)

  6. 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: • ;

  7. 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].

  8. 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.

  9. 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]

  10. 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.

  11. 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]

  12. The codes Lorentz - Lifted Codes • Alphabet: • Coordinates: • Parameter: degree • Message space: lines • Code: Evaluations of message on all of • And oh …

  13. 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.

  14. Generalizing: Lifted Codes Lorentz - Lifted Codes • Consider • extends • Preferably invariant under affine transformations of . • Lifted code • -dim. affine subspaces • Previous example:

  15. Properties of lifted codes Lorentz - Lifted Codes • Distance: • Local Decodability: • Same decoding algorithm as for RM codes. • is -LDC implies is -LDC. • Local Testability?

  16. 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.

  17. 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

  18. 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 ?

  19. Lucas’s theorem & Rate Lorentz - Lifted Codes Notation: if and ( and for all . Lucas’s Theorem: iff iff So given

  20. Binary addition etc. 0 0 0 0 0 msb lsb Lorentz - Lifted Codes

  21. Other lifted codes Lorentz - Lifted Codes • Best LCC with O(1) locality. • ; • -LCC; • Alternate codes for BGHMRS construction:

  22. 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]:

  23. 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).

  24. 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?

  25. Thank You Lorentz - Lifted Codes

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