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!