Locally Testable Codes and - PowerPoint Presentation

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Locally Testable Codes and Caylay Graphs

Parikshit

Gopalan

(MSR-SVC)

Salil

Vadhan

(Harvard)

Yuan Zhou (CMU)

Locally Testable CodesLocal tester for an [n, k, d]2 linear code

C

Queries few coordinates

Accepts

codewords

Rejects words far from the code with high probability

[BenSasson-Harsha-Raskhodnikova’05]

: A local tester is a distribution

D

on (low-weight) dual

codewords

Locally Testable Codes[Blum-Luby-Rubinfeld’90, Rubinfeld-Sudan’92, Freidl-Sudan’95]: (strong) tester for an [n, k, d]

2

code

Queries coordinates according to

D

onε-smooth: queries each coordinate w.p. ≤ εRejects words at distance d w.p. ≥ δdBy definition: must have δ≤ε; would like δ=Ω(ε)

Distance from

C

Pr

[Reject]

1

d/2

ε

.1

The price of locality?Asymptotically good regime#information bits k =

Ω

(n), distance d =

Ω

(n)

Are there asymptotically good 3-query LTCs?Existential question proposed by [Goldreich-Sudan’02]Best construction: n=k polylog(k), d = Ω(n) [Dinur’05]Rate-1 regime: let d be a large constant, ε=Θ(1/d), n∞

How large can k be for an [n, k, d]2 ε-smooth LTC?BCH: n-k = (d/2) log(n), but not locally testable

[BKSSZ’08]: n-k = log(n)log(d) from Reed-Muller

Can we have n-k = Od

(log(n))?

Caylay graphs on .

Graph

Vertices:

Edges:

Hypercube:

h

=

n

,

We are interested in

h

<

n

Definition.

S is

d-wise independent

if every subset T of S, where |T|<d, is linearly independent

Caylay graphs on .

Graph

Vertices:

Edges:

d-wise independent

:

Abelian

analogue of large girth

Cycles occur when edge labels sum to 0

always has

4-cycles

Caylay graphs on .

Graph

Vertices:

Edges:

d-wise independent

:

Abelian

analogue of large girth

Cycles occur when edge labels sum to 0

always has

4-cycles

non-trivial cycles have length at least d

Caylay graphs on .

Graph

Vertices:

Edges:

d-wise independent

:

Abelian

analogue of large girth

Cycles occur when edge labels sum to 0

always

has

4-cycles

non-trivial cycles have length at least d

(d/2)-neighborhood of any vertex is isomorphic to B(n, d/2), but the vertex set has dimension h << n

embeddings of graph

Embedding f: V(G)

 R

d

has distortion c if for every x, y

|f(x) – f(y)|1 ≤ dG(x, y) ≤ c|f(x) – f(y)|1 c1(G) = minimum distortion over all

embeddings

Our resultsTheorem. The following are equivalent

An [n, k, d]

2

code

C

with a tester of smoothness ε and soundness δA Cayley graph where |S| = n, S is d-wise independent, and the graph has an embedding of distortion ε/δCorollary. There exist asymptotically good strong LTCs iff there exists s.t.|S| = (1+Ω(1))h

S is Ω(h)-wise independentc1(G) = O(1)

Our resultsTheorem. The following are equivalent

An [n, k, d]

2

code

C

with a tester of smoothness ε and soundness δA Cayley graph where |S| = n, S is d-wise independent, and the graph has an embedding of distortion ε/δCorollary. There exist [n, n-Od(log n), d]2 strong LTCs iff there exists

s.t.|S| = 2Ωd(h)S is d-wise independentc1(G) = O(1)

Our resultsTheorem. [n, k, d]2 LTCs are equivalent to

Cayley

graphs on whose eigenvalue spectrum resembles the n-dimensional

ε

-noisy hypercube for

ε=1/dA converse to the result by [Barak-Gopalan-Håstad-Meka-Raghavendra-Steurer’12]

The correspondence

, |S|=n, S is d-wise

indep

.

[n, k, d]

2

code C: (n-k) x n parity check matrix [s1, s2, …,

sn]

Vertex set: F

2

n

/

C,Edge set: .

Claim.

Shortest path between and equals the shortest Hamming distance from (x – y) to a

codeword

.

To show:

the correspondence between

embeddings

and local testers.

Embeddings from testersGiven a tester distribution

D

on , each

a

~ D defines a cut on V(G) = F2n/C  an embeddingClaim. The embedding has distortion ε/δ

Proof. Given two nodes and

Testers from EmbeddingsGiven embedding distribution

D

on

If

D supported on linear functions, we’d be (essentially) done.Claim. There is a distribution D’ on linear functions with distortion as good as D.Proof sketch.Extend f to all points in

The Fourier expansion is supported on : When D samples f, D’ samples

w.p.

Applications[Khot-Naor’06]: If has distance

Ω

(n) and relative rate

Ω

(1), then c

1(G) = Ω(n) where G is the Caylay graph defined by C as described beforeProof. Suffices to lowerbound ε/δSince has distance Ω(n), we have ε

=Ω(1)Let t be the covering radius of C, we haveδ ≤ 1/t (since the rej

. prob. can be tδ)t =

Ω(n) (since has distance

Ω(n))Therefore ε/δ ≥

εt = Ω(n)

Future directionsCan we use this equivalence to prove better constructions (or better lower bounds) for LTCs?

Thanks!