380 research outputs found
Iterative Construction of Regular LDPC Codes from Independent Tree-Based Minimum Distance Bounds
An independent tree-based method for lower bounding the minimum distance of low-density parity-check (LDPC) codes is presented. This lower-bound is then used as the decision criterion during the iterative construction of regular LDPC codes. The new construction algorithm results in LDPC codes with greater girth and improved minimum-distance bounds when compared to regular LDPC codes constructed using the progressive edge-growth (PEG) construction and the approximate cycle extrinsic message degree (ACE)-constrained PEG construction. Simulation results of codes constructed with the new method show improved performance on the additive white Gaussian noise channel at moderate signal-to-noise ratios
Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights
We present a tree-based construction of LDPC codes that have minimum
pseudocodeword weight equal to or almost equal to the minimum distance, and
perform well with iterative decoding. The construction involves enumerating a
-regular tree for a fixed number of layers and employing a connection
algorithm based on permutations or mutually orthogonal Latin squares to close
the tree. Methods are presented for degrees and , for a
prime. One class corresponds to the well-known finite-geometry and finite
generalized quadrangle LDPC codes; the other codes presented are new. We also
present some bounds on pseudocodeword weight for -ary LDPC codes. Treating
these codes as -ary LDPC codes rather than binary LDPC codes improves their
rates, minimum distances, and pseudocodeword weights, thereby giving a new
importance to the finite geometry LDPC codes where .Comment: Submitted to Transactions on Information Theory. Submitted: Oct. 1,
2005; Revised: May 1, 2006, Nov. 25, 200
The Generalized Area Theorem and Some of its Consequences
There is a fundamental relationship between belief propagation and maximum a
posteriori decoding. The case of transmission over the binary erasure channel
was investigated in detail in a companion paper. This paper investigates the
extension to general memoryless channels (paying special attention to the
binary case). An area theorem for transmission over general memoryless channels
is introduced and some of its many consequences are discussed. We show that
this area theorem gives rise to an upper-bound on the maximum a posteriori
threshold for sparse graph codes. In situations where this bound is tight, the
extrinsic soft bit estimates delivered by the belief propagation decoder
coincide with the correct a posteriori probabilities above the maximum a
posteriori threshold. More generally, it is conjectured that the fundamental
relationship between the maximum a posteriori and the belief propagation
decoder which was observed for transmission over the binary erasure channel
carries over to the general case. We finally demonstrate that in order for the
design rate of an ensemble to approach the capacity under belief propagation
decoding the component codes have to be perfectly matched, a statement which is
well known for the special case of transmission over the binary erasure
channel.Comment: 27 pages, 46 ps figure
Diagnosis of weaknesses in modern error correction codes: a physics approach
One of the main obstacles to the wider use of the modern error-correction
codes is that, due to the complex behavior of their decoding algorithms, no
systematic method which would allow characterization of the Bit-Error-Rate
(BER) is known. This is especially true at the weak noise where many systems
operate and where coding performance is difficult to estimate because of the
diminishingly small number of errors. We show how the instanton method of
physics allows one to solve the problem of BER analysis in the weak noise range
by recasting it as a computationally tractable minimization problem.Comment: 9 pages, 8 figure
Windowed Decoding of Protograph-based LDPC Convolutional Codes over Erasure Channels
We consider a windowed decoding scheme for LDPC convolutional codes that is
based on the belief-propagation (BP) algorithm. We discuss the advantages of
this decoding scheme and identify certain characteristics of LDPC convolutional
code ensembles that exhibit good performance with the windowed decoder. We will
consider the performance of these ensembles and codes over erasure channels
with and without memory. We show that the structure of LDPC convolutional code
ensembles is suitable to obtain performance close to the theoretical limits
over the memoryless erasure channel, both for the BP decoder and windowed
decoding. However, the same structure imposes limitations on the performance
over erasure channels with memory.Comment: 18 pages, 9 figures, accepted for publication in the IEEE
Transactions on Information Theor
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
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