154 research outputs found
Query-Efficient Locally Decodable Codes of Subexponential Length
We develop the algebraic theory behind the constructions of Yekhanin (2008)
and Efremenko (2009), in an attempt to understand the ``algebraic niceness''
phenomenon in . We show that every integer ,
where , and are prime, possesses the same good algebraic property as
that allows savings in query complexity. We identify 50 numbers of this
form by computer search, which together with 511, are then applied to gain
improvements on query complexity via Itoh and Suzuki's composition method. More
precisely, we construct a -query LDC for every positive
integer and a -query
LDC for every integer , both of length , improving the
queries used by Efremenko (2009) and queries used by Itoh and
Suzuki (2010).
We also obtain new efficient private information retrieval (PIR) schemes from
the new query-efficient LDCs.Comment: to appear in Computational Complexit
New Constructions for Query-Efficient Locally Decodable Codes of Subexponential Length
A -locally decodable code
is an error-correcting code that encodes each message
to
and has the following property: For any such that
and each , the symbol
of can be recovered with probability at least by
a randomized decoding algorithm looking only at coordinates of .
The efficiency of a -locally decodable code is measured by the code length and the number of
queries. For any -query locally decodable code ,
the code length is conjectured to be exponential of , however, this was
disproved. Yekhanin [In Proc. of STOC, 2007] showed that there exists a 3-query
locally decodable code such that
assuming that the number of Mersenne primes is
infinite. For a 3-query locally decodable code ,
Efremenko [ECCC Report No.69, 2008] reduced the code length further to
, and also showed that for any
integer , there exists a -query locally decodable code such that and . In this paper, we present a query-efficient locally decodable
code and show that for any integer , there exists a -query locally
decodable code such that
and .Comment: 13 pages, 1 figure, 2 table
Three Query Locally Decodable Codes with Higher Correctness Require Exponential Length
Locally decodable codes are error correcting codes with the extra property that, in order to retrieve the correct value of just one position of the input with high probability, it is sufficient to read a small number of positions of the corresponding,
possibly corrupted codeword. A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length.
The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is for nonbinary codes, where an adversary is allowed to corrupt up to delta fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff, and it is below 1 - 3 delta.
We show that achieving slightly larger correctness (as a function of ) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes. Our results hold for linear codes over arbitrary finite fields and for binary nonlinear codes.
Considering larger number of queries, we obtain lower bounds for q-query codes for q>3, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms.
In addition, our results imply tradeoffs on the parameters of error correcting data structures
A Storage-Efficient and Robust Private Information Retrieval Scheme Allowing Few Servers
Since the concept of locally decodable codes was introduced by Katz and
Trevisan in 2000, it is well-known that information the-oretically secure
private information retrieval schemes can be built using locally decodable
codes. In this paper, we construct a Byzantine ro-bust PIR scheme using the
multiplicity codes introduced by Kopparty et al. Our main contributions are on
the one hand to avoid full replica-tion of the database on each server; this
significantly reduces the global redundancy. On the other hand, to have a much
lower locality in the PIR context than in the LDC context. This shows that
there exists two different notions: LDC-locality and PIR-locality. This is made
possible by exploiting geometric properties of multiplicity codes
2-Server PIR with sub-polynomial communication
A 2-server Private Information Retrieval (PIR) scheme allows a user to
retrieve the th bit of an -bit database replicated among two servers
(which do not communicate) while not revealing any information about to
either server. In this work we construct a 1-round 2-server PIR with total
communication cost . This improves over the
currently known 2-server protocols which require communication and
matches the communication cost of known 3-server PIR schemes. Our improvement
comes from reducing the number of servers in existing protocols, based on
Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing
these protocols in an algebraic way (using polynomial interpolation) and
extending them using partial derivatives
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
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