Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes Kenji Obata

Codes Error correcting code C : {0,1} n {0,1} m with decoding procedure A s.t. for y {0,1} m with d(y,C(x)) δm, A(y) = x

Locally Decodable Codes Weaken power of A: Can only look at a constant number q of input bits Weaken requirements: A need only recover a single given bit of x Can fail with some probability bounded away from ½ Study initiated by Katz and Trevisan [KT00]

Locally Decodable Codes Define a (q, δ, )-locally decodable code: A can make q queries (w.l.o.g. exactly q queries) For all x {0,1} n, all y {0,1} m with d(y, C(x)) δm, all inputs bits i 1,…, n A(y, i) = x i w/ probability ½ +

LDC Applications Direct: Scalable fault-tolerant information storage Indirect: Lower bounds for certain classes of private information retrieval schemes (more on this later)

Lower Bounds for LDCs [KT00] proved a general lower bound m n q/(q-1) (at best n 2, but known codes exponential) For 2-query linear LDCs Goldreich, Karloff, Schulman, Trevisan [GKST02] proved an exponential bound m 2 Ω(εδn)

Lower Bounds for LDCs Restriction to linear codes interesting, since known LDC constructions are linear But 2 Ω(εδn) not quite right: –Lower bound should increase arbitrarily as decoding probability 1 ( ε ½) –No matching construction

Lower Bounds for LDCs In this work, we prove that for 2-query linear LDCs, m 2 Ω(δ/(1-2ε)n) Optimal: There is an LDC construction matching this within a constant factor in the exponent

Techniques from [KT00] Fact: An LDC is also a smooth code (A queries each position w/ roughly the same probability) … so can study smooth codes Connects LDCs to information-theoretic PIR schemes: q queries q servers smoothness statistical indistinguishability

Techniques from [KT00] For i 1,…,n, define the recovery graph G i associated with C: Vertex set {1,…,m} (bits of the codeword) Edges are pairs (q 1, q 2 ) such that, conditioned on A querying q 1, q 2, A(C(x),i) outputs x i with prob > ½ Call these edges good edges (endpoints contain information about x i )

Techniques from [KT00]/[GKST02] Theorem: If C is (2, c, ε )-smooth, then G i contains a matching of size εm/c. Better to work with non-degenerate codes Each bit of the encoding depends on more than one bit of the message For linear codes, good edges are non-trivial linear combinations Fact: Any smooth code can be made non-degenerate (with constant loss in parameters).

Core Lemma [GKST02] Let q 1,…,q m be linear functions on {0,1} n s.t. for every i 1,…,n there is a set M i of at least γm disjoint pairs of indices j 1, j 2 such that x i = q j 1 (x) + q j 2 (x). Then m 2 γn.

Putting it all together… If C is a (2, c, )-smooth linear code, then (by reduction to non-degenerate code + existence of large matchings + core lemma), m 2 n/4c. If C is a (2, δ, )-locally decodable linear code, then (by LDC smooth reduction), m 2 δn/8.

Putting it all together… Summary: locally decodable smooth big matchings exponential size This work: locally decodable big matchings (skip smoothness reduction, argue directly about LDCs)

The Blocking Game Let G(V,E) be a graph on n vertices, w a prob distribution on E, X w an edge sampled according to w, S a subset of V Define the blocking probability β δ (G) as min w ( max |S|δn Pr (X w intersects S) )

The Blocking Game Want to characterize β δ (G) in terms of size of a maximum matching M(G), equivalently defect d(G) = n – 2M(G) Theorem: Let G be a graph with defect αn. Then β δ (G) min (δ/(1-α), 1).

The Blocking Game clique αnαn (1-α)n Define K(n,α) to be the edge- maximal graph on n vertices with defect αn: K1K1 K2K2

The Blocking Game Optimization on K(n,α) is a relaxation of optimization on any graph with defect αn If d(G) αn then β δ (G) β δ (K(n,α)) So, enough to think about K(n,α).

The Blocking Game Intuitively, best strategy for player 1 is to spread distribution as uniformly as possible A (λ 1,λ 2 )-symmetric dist: all edges in (K 1,K 2 ) have weight λ 1 all edges in (K 2,K 2 ) have weight λ 2 Lemma: (λ 1,λ 2 )-symmetric dist w s.t. β δ (K(n,α)) = max |S|δn Pr (X w intersects S).

The Blocking Game Claim: Let w 1,…,w k be dists s.t. max |S|δn Pr (X w i intersects S) = β δ (G). Then for any convex comb w = γ i w i max |S|δn Pr (X w intersects S) = β δ (G). Proof: For S V, |S| δn, intersection prob is γ i β δ (G) = β δ (G). So max |S| δn Pr (X w intersects S) β δ (G). But by defn of β δ (G), this must be β δ (G).

The Blocking Game Proof: Let w be any distribution optimizing β δ (G). If w does, then so does π(w) for π Aut(G) = Γ. By prior claim, so does w = (1/|Γ|) π Γ π(w). For e E, σ Γ, w(e) = (1/|Γ|) π Γ w(π(e)) = (1/|Γ|) π Γ w(πσ(e)) = w(σ(e)).. So, if e, e are in the same Γ-orbit, they have the same weight in w w is (λ 1,λ 2 )-symmetric.

The Blocking Game Claim: If w is (λ 1,λ 2 )-sym then S V, |S| δn s.t. Pr (X w intersects S) min (δ/(1-α), 1). Proof: If δ 1 – α then can cover every edge. Otherwise, set S = any δn vertices of K 2. Then Pr = δ ( 1/(1 - α) + ½ n 2 (1 - α – δ) λ 2 ) which, for δ < 1 - α, is at least δ/(1 - α) (optimized when λ 2 = 0).

The Blocking Game Theorem: Let G be a graph with defect αn. Then β δ (G) min (δ/(1-α), 1). Proof: β δ (G) β δ (K(n,α)). Blocking prob on K(n,α) is optimized by some (λ 1,λ 2 )-sym dist. For any such dist w, δn vertices blocking w with Pr min (δ/(1-α), 1).

Lower Bound for LDLCs Still need a degenerate non-degenerate reduction (this time, for LDCs instead of smooth codes) Theorem: Let C be a (2, δ, ε)-locally decodable linear code. Then, for large enough n, there exists a non-degenerate (2, δ/2.01, ε)-locally decodable linear code C : {0,1} n {0,1} 2m.

Lower Bound for LDLCs Theorem: Let C be a (2, δ, ε)-LDLC. Then, for large enough n, m 2 1/4.03 δ/(1-2ε) n. Proof: Make C non-degenerate Local decodability low blocking probability (at most ¼ - ½ ε) low defect (α 1 – (δ/2.01)/(1-2ε)) big matching (½ (δ/2.01)/(1-2ε) (2m) ) exponentially long encoding (m 2 (1/4.02) δ/(1-2ε)n – 1 )

Matching Upper Bound Hadamard code on {0,1} n y i = a i · x (a i runs through {0,1} n ) 2-query locally decodable Recovery graphs are perfect matchings on n-dim hypercube Success parameter ε = ½ - 2δ Can use concatenated Hadamard codes (Trevisan):

Matching Upper Bound Set c = (1-2ε)/4δ (can be shown that for feasible values of δ, ε, c 1). Divide input into c blocks of n/c bits, encode each block with Hadamard code on {0,1} n/c. Each block has a fraction cδ corrupt entries, so code has recovery parameter ½ - 2 (1-2ε)/4δ δ = ε Code has length (1-2ε)/4δ 2 4δ/(1-2ε)n

Conclusions There is a matching upper bound (concatenated Hadamard code) New results for 2-query non-linear codes (but using apparently completely different techniques) q > 2? –No analog to the core lemma for more queries –But blocking game analysis might generalize to useful properties other than matching size