163 research outputs found
A Message-Passing Algorithm for Counting Short Cycles in a Graph
A message-passing algorithm for counting short cycles in a graph is
presented. For bipartite graphs, which are of particular interest in coding,
the algorithm is capable of counting cycles of length g, g +2,..., 2g - 2,
where g is the girth of the graph. For a general (non-bipartite) graph, cycles
of length g; g + 1, ..., 2g - 1 can be counted. The algorithm is based on
performing integer additions and subtractions in the nodes of the graph and
passing extrinsic messages to adjacent nodes. The complexity of the proposed
algorithm grows as , where is the number of edges in the
graph. For sparse graphs, the proposed algorithm significantly outperforms the
existing algorithms in terms of computational complexity and memory
requirements.Comment: Submitted to IEEE Trans. Inform. Theory, April 21, 2010
How Fast Can Dense Codes Achieve the Min-Cut Capacity of Line Networks?
In this paper, we study the coding delay and the average coding delay of
random linear network codes (dense codes) over line networks with deterministic
regular and Poisson transmission schedules. We consider both lossless networks
and networks with Bernoulli losses. The upper bounds derived in this paper,
which are in some cases more general, and in some other cases tighter, than the
existing bounds, provide a more clear picture of the speed of convergence of
dense codes to the min-cut capacity of line networks.Comment: 15 pages, submitted to IEEE ISIT 201
From Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes
Cages, defined as regular graphs with minimum number of nodes for a given
girth, are well-studied in graph theory. Trapping sets are graphical structures
responsible for error floor of low-density parity-check (LDPC) codes, and are
well investigated in coding theory. In this paper, we make connections between
cages and trapping sets. In particular, starting from a cage (or a modified
cage), we construct a trapping set in multiple steps. Based on the connection
between cages and trapping sets, we then use the available results in graph
theory on cages and derive tight upper bounds on the size of the smallest
trapping sets for variable-regular LDPC codes with a given variable degree and
girth. The derived upper bounds in many cases meet the best known lower bounds
and thus provide the actual size of the smallest trapping sets. Considering
that non-zero codewords are a special case of trapping sets, we also derive
tight upper bounds on the minimum weight of such codewords, i.e., the minimum
distance, of variable-regular LDPC codes as a function of variable degree and
girth
Ultra Low-Complexity Detection of Spectrum Holes in Compressed Wideband Spectrum Sensing
Wideband spectrum sensing is a significant challenge in cognitive radios
(CRs) due to requiring very high-speed analog- to-digital converters (ADCs),
operating at or above the Nyquist rate. Here, we propose a very low-complexity
zero-block detection scheme that can detect a large fraction of spectrum holes
from the sub-Nyquist samples, even when the undersampling ratio is very small.
The scheme is based on a block sparse sensing matrix, which is implemented
through the design of a novel analog-to- information converter (AIC). The
proposed scheme identifies some measurements as being zero and then verifies
the sub-channels associated with them as being vacant. Analytical and
simulation results are presented that demonstrate the effectiveness of the
proposed method in reliable detection of spectrum holes with complexity much
lower than existing schemes. This work also introduces a new paradigm in
compressed sensing where one is interested in reliable detection of (some of
the) zero blocks rather than the recovery of the whole block sparse signal.Comment: 7 pages, 5 figure
Characterization and Efficient Search of Non-Elementary Trapping Sets of LDPC Codes with Applications to Stopping Sets
In this paper, we propose a characterization for non-elementary trapping sets
(NETSs) of low-density parity-check (LDPC) codes. The characterization is based
on viewing a NETS as a hierarchy of embedded graphs starting from an ETS. The
characterization corresponds to an efficient search algorithm that under
certain conditions is exhaustive. As an application of the proposed
characterization/search, we obtain lower and upper bounds on the stopping
distance of LDPC codes.
We examine a large number of regular and irregular LDPC codes, and
demonstrate the efficiency and versatility of our technique in finding lower
and upper bounds on, and in many cases the exact value of, . Finding
, or establishing search-based lower or upper bounds, for many of the
examined codes are out of the reach of any existing algorithm
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