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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, FreidlSudan’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))? h 2 Caylay graphs on F . Graph Cay(F2h, S) – S = {s1, s2,… , sn } Í F2h – Vertices: F2h – Edges: {(x, x + si ) : x Î F2h,i Î [n]} Hypercube: h = n, S = {e1, e2,… , eh } We are interested in h < n • Definition. S is d-wise independent if every subset T of S, where |T|<d, is linearly independent h 2 Caylay graphs on F . s1 s2 s2 s1 Graph Cay(F2h, S) – S = {s1, s2,… , sn } Í F2h – Vertices: F2h – Edges: {(x, x + si ) : x Î F2h,i Î [n]} d-wise independent: Abelian analogue of large girth • Cycles occur when edge labels sum to 0 • Cay(F2h, S) always has 4-cycles h 2 Caylay graphs on F . Graph Cay(F2h, S) – S = {s1, s2,… , sn } Í F2h – Vertices: F2h – Edges: {(x, x + si ) : x Î F2h,i Î [n]} d-wise independent: Abelian analogue of large girth • Cycles occur when edge labels sum to 0 • Cay(F2h, S) always has 4-cycles • non-trivial cycles have length at least d h 2 Caylay graphs on F . Graph Cay(F2h, S) – S = {s1, s2,… , sn } Í F2h – Vertices: F2h – Edges: {(x, x + si ) : x Î F2h,i Î [n]} d-wise independent: Abelian analogue of large girth • Cycles occur when edge labels sum to 0 • Cay(F2h, S) 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 ℓ1 - 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 Cay(F2n-k , S) where |S| = n, S is dwise independent, and the graph has an ℓ1 embedding of distortion ε/δ • Corollary. There exist asymptotically good strong LTCs h iff there exists G = Cay(Fs.t. 2 , S) – |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 Cay(F2n-k , S) where |S| = n, S is dwise independent, and the graph has an ℓ1 embedding of distortion ε/δ • Corollary. There exist [n, n-Od(log n), d]2 strong LTCs iff there exists G = Cay(F2h , S) s.t. – |S| = 2Ωd(h) – S is d-wise independent – c1(G) = O(1) The correspondence Cay(F2n-k , S), |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: ( x, x + e1 ), ( x, x + e2 ),… , ( x, x + en ) . Claim. Shortest path between x and y equals the shortest Hamming distance from (x – y) to a codeword. To show: the correspondence between ℓ1 -embeddings and local testers. Embeddings from testers • Given a tester distribution D on , each a ~ D defines a cut on V(G) = F2n/C anℓ1 - embedding • Claim. The embedding has distortion ε/δ • Proof. Given two nodes x and y dist G (x, y) = wt(x - y) d × wt(x - y) £ Pr[(x - y) is rejected] £ e × wt(x - y) 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 F2n – The Fourier expansion is supported on : – When D samples f, D’ samples ( ) c aw.p. fˆa 2 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!