143 research outputs found
Average Stopping Set Weight Distribution of Redundant Random Matrix Ensembles
In this paper, redundant random matrix ensembles (abbreviated as redundant
random ensembles) are defined and their stopping set (SS) weight distributions
are analyzed. A redundant random ensemble consists of a set of binary matrices
with linearly dependent rows. These linearly dependent rows (redundant rows)
significantly reduce the number of stopping sets of small size. An upper and
lower bound on the average SS weight distribution of the redundant random
ensembles are shown. From these bounds, the trade-off between the number of
redundant rows (corresponding to decoding complexity of BP on BEC) and the
critical exponent of the asymptotic growth rate of SS weight distribution
(corresponding to decoding performance) can be derived. It is shown that, in
some cases, a dense matrix with linearly dependent rows yields asymptotically
(i.e., in the regime of small erasure probability) better performance than
regular LDPC matrices with comparable parameters.Comment: 14 pages, 7 figures, Conference version to appear at the 2007 IEEE
International Symposium on Information Theory, Nice, France, June 200
On the Asymptotic Weight and Stopping Set Distribution of Regular LDPC Ensembles
We estimate the variance of weight and stopping set distribution of regular
LDPC ensembles. Using this estimate and the second moment method we obtain
bounds on the probability that a randomly chosen code from regular LDPC
ensemble has its weight distribution and stopping set distribution close to
respective ensemble averages. We are able to show that a large fraction of
total number of codes have their weight and stopping set distribution close to
the average.Comment: Submitted to IEEE Trans on Information Theory (revised version).
Minor comments from reviewers addresse
Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation
This paper analyzes the asymptotic exponent of the weight spectrum for
irregular doubly-generalized LDPC (D-GLDPC) codes. In the process, an efficient
numerical technique for its evaluation is presented, involving the solution of
a 4 x 4 system of polynomial equations. The expression is consistent with
previous results, including the case where the normalized weight or stopping
set size tends to zero. The spectral shape is shown to admit a particularly
simple form in the special case where all variable nodes are repetition codes
of the same degree, a case which includes Tanner codes; for this case it is
also shown how certain symmetry properties of the local weight distribution at
the CNs induce a symmetry in the overall weight spectral shape function.
Finally, using these new results, weight and stopping set size spectral shapes
are evaluated for some example generalized and doubly-generalized LDPC code
ensembles.Comment: 17 pages, 6 figures. To appear in IEEE Transactions on Information
Theor
Spectral Shape of Check-Hybrid GLDPC Codes
This paper analyzes the asymptotic exponent of both the weight spectrum and
the stopping set size spectrum for a class of generalized low-density
parity-check (GLDPC) codes. Specifically, all variable nodes (VNs) are assumed
to have the same degree (regular VN set), while the check node (CN) set is
assumed to be composed of a mixture of different linear block codes (hybrid CN
set). A simple expression for the exponent (which is also referred to as the
growth rate or the spectral shape) is developed. This expression is consistent
with previous results, including the case where the normalized weight or
stopping set size tends to zero. Furthermore, it is shown how certain symmetry
properties of the local weight distribution at the CNs induce a symmetry in the
overall weight spectral shape function.Comment: 6 pages, 3 figures. Presented at the IEEE ICC 2010, Cape Town, South
Africa. A minor typo in equation (9) has been correcte
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
Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance
Parameters of LDPC codes, such as minimum distance, stopping distance,
stopping redundancy, girth of the Tanner graph, and their influence on the
frame error rate performance of the BP, ML and near-ML decoding over a BEC and
an AWGN channel are studied. Both random and structured LDPC codes are
considered. In particular, the BP decoding is applied to the code parity-check
matrices with an increasing number of redundant rows, and the convergence of
the performance to that of the ML decoding is analyzed. A comparison of the
simulated BP, ML, and near-ML performance with the improved theoretical bounds
on the error probability based on the exact weight spectrum coefficients and
the exact stopping size spectrum coefficients is presented. It is observed that
decoding performance very close to the ML decoding performance can be achieved
with a relatively small number of redundant rows for some codes, for both the
BEC and the AWGN channels
Growth Rate of the Weight Distribution of Doubly-Generalized LDPC Codes: General Case and Efficient Evaluation
The growth rate of the weight distribution of irregular doubly-generalized
LDPC (D-GLDPC) codes is developed and in the process, a new efficient numerical
technique for its evaluation is presented. The solution involves simultaneous
solution of a 4 x 4 system of polynomial equations. This represents the first
efficient numerical technique for exact evaluation of the growth rate, even for
LDPC codes. The technique is applied to two example D-GLDPC code ensembles.Comment: 6 pages, 1 figure. Proc. IEEE Globecom 2009, Hawaii, USA, November 30
- December 4, 200
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