Locally Testable Codes and Caylay Graphs

1. Locally Testable Codes and Caylay Graphs ParikshitGopalan(MSR-SVC) SalilVadhan (Harvard) Yuan Zhou (CMU)

2. 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

3. 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) Pr[Reject] .1 ε Distance from C 1 d/2

4. 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]2ε-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))?

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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)

11. 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)

12. The correspondence , |S|=n, S is d-wise indep. [n, k, d]2code 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.

13. Embeddings from testers • Given a tester distribution Don , each a ~Ddefines a cut on V(G) = F2n/C  an embedding • Claim. The embedding has distortion ε/δ • Proof. Given two nodes and

14. Testers from Embeddings • Given embedding distribution Don 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.

15. 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 lowerboundε/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)

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

17. Thanks!