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] 1d/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 = O d (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 ≤ d G (x, y) ≤ c|f(x) – f(y)| 1 c 1 (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 – c 1 (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-O d (log n), d] 2 strong LTCs iff there exists s.t. – |S| = 2 Ω d (h) – S is d-wise independent – c 1 (G) = O(1)

Our results Theorem. [n, k, d] 2 LTCs are equivalent to Cayley graphs on whose eigenvalue spectrum resembles the n-dimensional ε-noisy hypercube for ε=1/d – A 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 [s 1, s 2, …, s n ] 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 testers Given a tester distribution D on, each a ~ D defines a cut on V(G) = F 2 n / C  an embedding Claim. The embedding has distortion ε/δ Proof. Given two nodes and

Testers from Embeddings Given 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 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!