Optimal Lower Bounds for 2Query Locally Decodable Linear Codes

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Optimal Lower Bounds for2-Query Locally
Decodable Linear Codes

• Kenji Obata

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Codes

• Error correcting code
• C 0,1n ? 0,1m
• with decoding procedure A s.t.
• for y ? 0,1m with d(y,C(x)) dm,
• A(y) x

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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

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Locally Decodable Codes

• Define a (q, d, ?)-locally decodable code
• A can make q queries (w.l.o.g. exactly q
  queries)
• For all x ? 0,1n, all y ? 0,1m with d(y,
  C(x)) dm, all inputs bits i ? 1,, n
• A(y, i) xi w/ probability ½ ?

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LDC Applications

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

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Lower Bounds for LDCs

• KT00 proved a general lower bound
• m nq/(q-1)
• (at best n2, but known codes exponential)
• For 2-query linear LDCsGoldreich, Karloff,
  Schulman, Trevisan GKST02 proved an exponential
  bound
• m 2O(edn)

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Lower Bounds for LDCs

• Restriction to linear codes interesting, since
  known LDC constructions are linear
• But 2O(edn) not quite right
• Lower bound should increase arbitrarily as
  decoding probability ? 1 (e ? ½)
• No matching construction

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Lower Bounds for LDCs

• In this work, we prove that for 2-query linear
  LDCs,
• m 2O(d/(1-2e)n)
• Optimal There is an LDC construction matching
  this within a constant factor in the exponent

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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

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Techniques from KT00

• For i ? 1,,n, define the recovery graph Gi
  associated with C
• Vertex set 1,,m (bits of the codeword)
• Edges are pairs (q1, q2) such that, conditioned
  on A querying q1, q2,
• A(C(x),i) outputs xi with prob gt ½
• Call these edges good edges (endpoints contain
  information about xi)

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Techniques from KT00/GKST02

• Theorem If C is (2, c, e)-smooth, then Gi
  contains a matching of size em/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).

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Core Lemma GKST02

• Let q1,,qm be linear functions on 0,1n s.t.
  for every i ? 1,,nthere is a set Mi of at least
  ?m disjoint pairs of indices j1, j2 such that
• xi qj1(x) qj2(x).
• Then m 2?n.

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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, d, ?)-locally decodable linear
  code, then (by LDC ? smooth reduction),
• m 2?dn/8.

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Putting it all together

• Summarylocally decodable ? smooth ?big
  matchings ? exponential size
• This worklocally decodable ? big matchings
• (skip smoothness reduction, argue directly about
  LDCs)

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The Blocking Game

• Let G(V,E) be a graph on n vertices, w a prob
  distribution on E, Xw an edge sampled according
  to w, S a subset of V
• Define the blocking probability ßd(G) as
• minw (maxSdn Pr (Xw intersects S))

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The Blocking Game

• Want to characterize ßd(G) in terms of size of a
  maximum matching M(G), equivalently defect d(G)
  n 2M(G)
• Theorem Let G be a graph withdefect an. Then
• ßd(G) min (d/(1-a), 1).

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The Blocking Game
Define K(n,a) to be the edge-maximal graph on n
vertices with defect an
clique
(1-a)n
an
K1
K2
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The Blocking Game

• Optimization on K(n,a) is a relaxation of
  optimization on any graph with defect an
• If d(G) an then
• ßd(G) ßd(K(n,a))
• So, enough to think about K(n,a).

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The Blocking Game

• Intuitively, best strategy for player 1 is to
  spread distribution as uniformly as possible
• A (?1,?2)-symmetric dist
• all edges in (K1,K2) have weight ?1
• all edges in (K2,K2) have weight ?2
• Lemma ? (?1,?2)-symmetric dist w s.t.
• ßd(K(n,a)) maxSdn Pr (Xw intersects S).

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The Blocking Game

• Claim Let w1,,wk be dists s.t.
• maxSdn Pr (Xwi intersects S) ßd(G).
• Then for any convex comb w ? ?i wi
• maxSdn Pr (Xw intersects S) ßd(G).
• Proof For S?? V, S dn, intersection prob is
  ? ?i ßd(G) ßd(G). So
• maxS dn Pr (Xw intersects S) ßd(G).
• But by defn of ßd(G), this must be ßd(G).

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The Blocking Game

• Proof Let w be any distribution optimizing
  ßd(G). If w does, then so does p(w) for p ?
  Aut(G) G. By prior claim, so does
• w (1/G) ?p?G p(w).
• For e?E, s?G,
• w(e) (1/G) ?p?G w(p(e))
• (1/G) ?p?G w(ps(e))
• w(s(e)). .
• So, if e, e are in the same G-orbit, they have
  the same weight in w ? w is (?1,?2)-symmetric.

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The Blocking Game

• Claim If w is (?1,?2)-sym then ? S ? V,S
  dn s.t.
• Pr (Xw intersects S) min (d/(1-a), 1).
• Proof If d 1 a then can cover every edge.
  Otherwise, set S any dn vertices of K2. Then
• Pr d (1/(1 - a) ½ n2 (1 - a d) ?2)
• which, for d lt 1 - a, is at least
• d/(1 - a)
• (optimized when ?2 0).

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The Blocking Game

• Theorem Let G be a graph withdefect an. Then
• ßd(G) min (d/(1-a), 1).
• Proof ßd(G) ßd(K(n,a)). Blocking prob on
  K(n,a) is optimized by some (?1,?2)-sym dist.
  For any such dist w, ? dn vertices blocking w
  with Pr min (d/(1-a), 1).

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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, d, e)-locally decodable
  linear code. Then, for large enough n, there
  exists a non-degenerate (2, d/2.01, e)-locally
  decodable linear code
• C 0,1n ? 0,12m.

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Lower Bound for LDLCs

• Theorem Let C be a (2, d, e)-LDLC. Then, for
  large enough n,
• m 21/4.03 d/(1-2e) n.
• Proof
• Make C non-degenerate
• Local decodability
• low blocking probability (at most ¼ - ½ e)
• low defect (a 1 (d/2.01)/(1-2e))
• big matching (½ (d/2.01)/(1-2e) (2m) )
• exponentially long encoding (m 2(1/4.02)
  d/(1-2e)n 1)

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Matching Upper Bound

• Hadamard code on 0,1n
• yi ai x (ai runs through 0,1n)
• 2-query locally decodable
• Recovery graphs are perfect matchings onn-dim
  hypercube
• Success parameter e ½ - 2d
• Can use concatenated Hadamard codes (Trevisan)

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Matching Upper Bound

• Set c (1-2e)/4d (can be shown that for feasible
  values of d, e, c 1).
• Divide input into c blocks of n/c bits, encode
  each block with Hadamard code on 0,1n/c.
• Each block has a fraction cd corrupt entries,
  so code has recovery parameter
• ½ - 2 (1-2e)/4d d e
• Code has length
• (1-2e)/4d 24d/(1-2e)n

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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 gt 2?
• No analog to the core lemma for more queries
• But blocking game analysis might generalize to
  useful properties other than matching size