146 research outputs found
Exponential Lower Bound for 2-Query Locally Decodable Codes via a Quantum Argument
A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in
such a way that one can recover any bit x_i from a corrupted codeword by
querying only a few bits of that word. We use a quantum argument to prove that
LDCs with 2 classical queries need exponential length: m=2^{Omega(n)}.
Previously this was known only for linear codes (Goldreich et al. 02). Our
proof shows that a 2-query LDC can be decoded with only 1 quantum query, and
then proves an exponential lower bound for such 1-query locally
quantum-decodable codes. We also show that q quantum queries allow more
succinct LDCs than the best known LDCs with q classical queries. Finally, we
give new classical lower bounds and quantum upper bounds for the setting of
private information retrieval. In particular, we exhibit a quantum 2-server PIR
scheme with O(n^{3/10}) qubits of communication, improving upon the O(n^{1/3})
bits of communication of the best known classical 2-server PIR.Comment: 16 pages Latex. 2nd version: title changed, large parts rewritten,
some results added or improve
Locally Decodable Quantum Codes
We study a quantum analogue of locally decodable error-correcting codes. A
q-query locally decodable quantum code encodes n classical bits in an m-qubit
state, in such a way that each of the encoded bits can be recovered with high
probability by a measurement on at most q qubits of the quantum code, even if a
constant fraction of its qubits have been corrupted adversarially. We show that
such a quantum code can be transformed into a classical q-query locally
decodable code of the same length that can be decoded well on average (albeit
with smaller success probability and noise-tolerance). This shows, roughly
speaking, that q-query quantum codes are not significantly better than q-query
classical codes, at least for constant or small q.Comment: 15 pages, LaTe
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval
We prove new lower bounds for locally decodable codes and private information
retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit
alphabet, where the decoder only uses b bits of each queried position of the
codeword, needs code length m = exp(Omega(n/(2^b Sum_{i=0}^b {l choose i})))
Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries,
where the user only needs b bits from each of the two l-bit answers, unknown to
the servers, satisfies t = Omega(n/(2^b Sum_{i=0}^b {l choose i})). This
implies that several known PIR schemes are close to optimal. Our results
generalize those of Goldreich et al. who proved roughly the same bounds for
linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf, our classical
lower bounds are proved using quantum computational techniques. In particular,
we give a tight analysis of how well a 2-input function can be computed from a
quantum superposition of both inputs.Comment: 12 pages LaTeX, To appear in ICALP '0
Error-Correcting Data Structures
We study data structures in the presence of adversarial noise. We want to
encode a given object in a succinct data structure that enables us to
efficiently answer specific queries about the object, even if the data
structure has been corrupted by a constant fraction of errors. This new model
is the common generalization of (static) data structures and locally decodable
error-correcting codes. The main issue is the tradeoff between the space used
by the data structure and the time (number of probes) needed to answer a query
about the encoded object. We prove a number of upper and lower bounds on
various natural error-correcting data structure problems. In particular, we
show that the optimal length of error-correcting data structures for the
Membership problem (where we want to store subsets of size s from a universe of
size n) is closely related to the optimal length of locally decodable codes for
s-bit strings.Comment: 15 pages LaTeX; an abridged version will appear in the Proceedings of
the STACS 2009 conferenc
Locally decodable codes and the failure of cotype for projective tensor products
It is shown that for every there exists a Banach space
of finite cotype such that the projective tensor product \ell_p\tp X fails to
have finite cotype. More generally, if satisfy
then
\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a
proved via a connection to the theory of locally decodable codes
Combinatorial Lower Bounds for 3-Query LDCs
A code is called a -query locally decodable code (LDC) if there is a
randomized decoding algorithm that, given an index and a received word
close to an encoding of a message , outputs by querying only at most
coordinates of . Understanding the tradeoffs between the dimension,
length and query complexity of LDCs is a fascinating and unresolved research
challenge. In particular, for -query binary LDCs of dimension and length
, the best known bounds are: .
In this work, we take a second look at binary -query LDCs. We investigate
a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query
LDCs. We prove an upper bound on the number of edges in these hypergraphs,
reproducing the known lower bound of for the length of
strong -query LDCs. In contrast to previous work, our techniques are purely
combinatorial and do not rely on a direct reduction to -query LDCs, opening
up a potentially different approach to analyzing 3-query LDCs.Comment: 10 page
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