144,483 research outputs found
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
Capacity of wireless erasure networks
In this paper, a special class of wireless networks, called wireless erasure networks, is considered. In these networks, each node is connected to a set of nodes by possibly correlated erasure channels. The network model incorporates the broadcast nature of the wireless environment by requiring each node to send the same signal on all outgoing channels. However, we assume there is no interference in reception. Such models are therefore appropriate for wireless networks where all information transmission is packetized and where some mechanism for interference avoidance is already built in. This paper looks at multicast problems over these networks. The capacity under the assumption that erasure locations on all the links of the network are provided to the destinations is obtained. It turns out that the capacity region has a nice max-flow min-cut interpretation. The definition of cut-capacity in these networks incorporates the broadcast property of the wireless medium. It is further shown that linear coding at nodes in the network suffices to achieve the capacity region. Finally, the performance of different coding schemes in these networks when no side information is available to the destinations is analyzed
The Three Node Wireless Network: Achievable Rates and Cooperation Strategies
We consider a wireless network composed of three nodes and limited by the
half-duplex and total power constraints. This formulation encompasses many of
the special cases studied in the literature and allows for capturing the common
features shared by them. Here, we focus on three special cases, namely 1) Relay
Channel, 2) Multicast Channel, and 3) Conference Channel. These special cases
are judicially chosen to reflect varying degrees of complexity while
highlighting the common ground shared by the different variants of the three
node wireless network. For the relay channel, we propose a new cooperation
scheme that exploits the wireless feedback gain. This scheme combines the
benefits of decode-and-forward and compress-and-forward strategies and avoids
the idealistic feedback assumption adopted in earlier works. Our analysis of
the achievable rate of this scheme reveals the diminishing feedback gain at
both the low and high signal-to-noise ratio regimes. Inspired by the proposed
feedback strategy, we identify a greedy cooperation framework applicable to
both the multicast and conference channels. Our performance analysis reveals
several nice properties of the proposed greedy approach and the central role of
cooperative source-channel coding in exploiting the receiver side information
in the wireless network setting. Our proofs for the cooperative multicast with
side-information rely on novel nested and independent binning encoders along
with a list decoder.Comment: 52 page
Coding Strategies for Noise-Free Relay Cascades with Half-Duplex Constraint
Two types of noise-free relay cascades are investigated. Networks where a
source communicates with a distant receiver via a cascade of half-duplex
constrained relays, and networks where not only the source but also a single
relay node intends to transmit information to the same destination. We
introduce two relay channel models, capturing the half-duplex constraint, and
within the framework of these models capacity is determined for the first
network type. It turns out that capacity is significantly higher than the rates
which are achievable with a straightforward time-sharing approach. A capacity
achieving coding strategy is presented based on allocating the transmit and
receive time slots of a node in dependence of the node's previously received
data. For the networks of the second type, an upper bound to the rate region is
derived from the cut-set bound. Further, achievability of the cut-set bound in
the single relay case is shown given that the source rate exceeds a certain
minimum value.Comment: Proceedings of the 2008 IEEE International Symposium on Information
Theory, Toronto, ON, Canada, July 6 - 11, 200
Shannon Information and Kolmogorov Complexity
We compare the elementary theories of Shannon information and Kolmogorov
complexity, the extent to which they have a common purpose, and where they are
fundamentally different. We discuss and relate the basic notions of both
theories: Shannon entropy versus Kolmogorov complexity, the relation of both to
universal coding, Shannon mutual information versus Kolmogorov (`algorithmic')
mutual information, probabilistic sufficient statistic versus algorithmic
sufficient statistic (related to lossy compression in the Shannon theory versus
meaningful information in the Kolmogorov theory), and rate distortion theory
versus Kolmogorov's structure function. Part of the material has appeared in
print before, scattered through various publications, but this is the first
comprehensive systematic comparison. The last mentioned relations are new.Comment: Survey, LaTeX 54 pages, 3 figures, Submitted to IEEE Trans
Information Theor
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