2,242 research outputs found
Codes for Graph Erasures
Motivated by systems where the information is represented by a graph, such as
neural networks, associative memories, and distributed systems, we present in
this work a new class of codes, called codes over graphs. Under this paradigm,
the information is stored on the edges of an undirected graph, and a code over
graphs is a set of graphs. A node failure is the event where all edges in the
neighborhood of the failed node have been erased. We say that a code over
graphs can tolerate node failures if it can correct the erased edges of
any failed nodes in the graph. While the construction of such codes can
be easily accomplished by MDS codes, their field size has to be at least
, when is the number of nodes in the graph. In this work we present
several constructions of codes over graphs with smaller field size. In
particular, we present optimal codes over graphs correcting two node failures
over the binary field, when the number of nodes in the graph is a prime number.
We also present a construction of codes over graphs correcting node
failures for all over a field of size at least , and show how
to improve this construction for optimal codes when .Comment: To appear in IEEE International Symposium on Information Theor
Tree Codes Improve Convergence Rate of Consensus Over Erasure Channels
We study the problem of achieving average consensus between a group of agents
over a network with erasure links. In the context of consensus problems, the
unreliability of communication links between nodes has been traditionally
modeled by allowing the underlying graph to vary with time. In other words,
depending on the realization of the link erasures, the underlying graph at each
time instant is assumed to be a subgraph of the original graph. Implicit in
this model is the assumption that the erasures are symmetric: if at time t the
packet from node i to node j is dropped, the same is true for the packet
transmitted from node j to node i. However, in practical wireless communication
systems this assumption is unreasonable and, due to the lack of symmetry,
standard averaging protocols cannot guarantee that the network will reach
consensus to the true average. In this paper we explore the use of channel
coding to improve the performance of consensus algorithms. For symmetric
erasures, we show that, for certain ranges of the system parameters, repetition
codes can speed up the convergence rate. For asymmetric erasures we show that
tree codes (which have recently been designed for erasure channels) can be used
to simulate the performance of the original "unerased" graph. Thus, unlike
conventional consensus methods, we can guarantee convergence to the average in
the asymmetric case. The price is a slowdown in the convergence rate, relative
to the unerased network, which is still often faster than the convergence rate
of conventional consensus algorithms over noisy links
Coding with Encoding Uncertainty
We study the channel coding problem when errors and uncertainty occur in the
encoding process. For simplicity we assume the channel between the encoder and
the decoder is perfect. Focusing on linear block codes, we model the encoding
uncertainty as erasures on the edges in the factor graph of the encoder
generator matrix. We first take a worst-case approach and find the maximum
tolerable number of erasures for perfect error correction. Next, we take a
probabilistic approach and derive a sufficient condition on the rate of a set
of codes, such that decoding error probability vanishes as blocklength tends to
infinity. In both scenarios, due to the inherent asymmetry of the problem, we
derive the results from first principles, which indicates that robustness to
encoding errors requires new properties of codes different from classical
properties.Comment: 12 pages; a shorter version of this work will appear in the
proceedings of ISIT 201
Cooperative Local Repair in Distributed Storage
Erasure-correcting codes, that support local repair of codeword symbols, have
attracted substantial attention recently for their application in distributed
storage systems. This paper investigates a generalization of the usual locally
repairable codes. In particular, this paper studies a class of codes with the
following property: any small set of codeword symbols can be reconstructed
(repaired) from a small number of other symbols. This is referred to as
cooperative local repair. The main contribution of this paper is bounds on the
trade-off of the minimum distance and the dimension of such codes, as well as
explicit constructions of families of codes that enable cooperative local
repair. Some other results regarding cooperative local repair are also
presented, including an analysis for the well-known Hadamard/Simplex codes.Comment: Fixed some minor issues in Theorem 1, EURASIP Journal on Advances in
Signal Processing, December 201
Linear-time list recovery of high-rate expander codes
We show that expander codes, when properly instantiated, are high-rate list
recoverable codes with linear-time list recovery algorithms. List recoverable
codes have been useful recently in constructing efficiently list-decodable
codes, as well as explicit constructions of matrices for compressive sensing
and group testing. Previous list recoverable codes with linear-time decoding
algorithms have all had rate at most 1/2; in contrast, our codes can have rate
for any . We can plug our high-rate codes into a
construction of Meir (2014) to obtain linear-time list recoverable codes of
arbitrary rates, which approach the optimal trade-off between the number of
non-trivial lists provided and the rate of the code. While list-recovery is
interesting on its own, our primary motivation is applications to
list-decoding. A slight strengthening of our result would implies linear-time
and optimally list-decodable codes for all rates, and our work is a step in the
direction of solving this important problem
- …