Locally Testable Codes and

Caylay Graphs

Parikshit Gopalan (MSR-SVC)Salil Vadhan (Harvard)

Yuan Zhou (CMU)

Locally Testable Codes

• Local 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 on– ε-smooth: queries each coordinate w.p. ≤ ε– Rejects words at distance d w.p. ≥ δd

• By 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) Rd 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 results• Theorem. The following are equivalent– An [n, k, d]2 code C with a tester of smoothness ε

and soundness δ– A Cayley graph where |S| = n, S is d-

wise independent, and the graph has an embedding of distortion ε/δ

• Corollary. There exist asymptotically good strong LTCs iff there exists s.t.– |S| = (1+Ω(1))h– S is Ω(h)-wise independent– c1(G) = O(1)

Our results• Theorem. The following are equivalent– An [n, k, d]2 code C with a tester of smoothness ε

and soundness δ– A Cayley graph where |S| = n, S is d-

wise independent, and the graph has an embedding of distortion ε/δ

• Corollary. There exist [n, n-Od(log n), d]2 strong LTCs iff there exists s.t.– |S| = 2Ωd(h)

– S is d-wise independent– c1(G) = O(1)

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: F2n/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 testers

• Given a tester distribution D on , each a ~ D defines a cut on V(G) = F2

n/C an embedding

• Claim. The embedding has distortion ε/δ• Proof. Given two nodes and

Testers from Embeddings• Given embedding distribution D onIf 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 c1(G) = Ω(n) where G is the Caylay graph defined by C as described before

• Proof. Suffices to lowerbound ε/δ– Since has distance Ω(n), we have ε=Ω(1)– Let t be the covering radius of C, we have• δ ≤ 1/t (since the rej. prob. can be tδ)• t = Ω(n) (since has distance Ω(n))

– Therefore ε/δ ≥ εt = Ω(n)

Future directions

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

Thanks!