Locally Testable Codes and Caylay Graphs Parikshit Gopalan
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 • [Ben. Sasson-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) Pr[Reject]. 1 ε 1 d/2 Distance from C
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 ≤ 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 dwise 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 dwise 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 – c 1(G) = O(1)
The correspondence , |S|=n, S is d-wise indep. Edge set: [n, k, d]2 code C: (n-k) x n parity check matrix [s 1, s 2, …, sn] Vertex set: F 2 n/C, . Claim. Shortest path between and equals the shortest Hamming distance from (x – y) to a codeword. To show: the correspondence between and local testers. embeddings
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!
- Slides: 17