63 research outputs found
On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes
In this paper, an expression for the asymptotic growth rate of the number of
small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC)
codes is derived. The expression is compact and generalizes existing results
for LDPC and generalized LDPC (GLDPC) codes. Assuming that there exist check
and variable nodes with minimum distance 2, it is shown that the growth rate
depends only on these nodes. An important connection between this new result
and the stability condition of D-GLDPC codes over the BEC is highlighted. Such
a connection, previously observed for LDPC and GLDPC codes, is now extended to
the case of D-GLDPC codes.Comment: 10 pages, 1 figure, presented at the 46th Annual Allerton Conference
on Communication, Control and Computing (this version includes additional
appendix
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
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
Minimum Distance Distribution of Irregular Generalized LDPC Code Ensembles
In this paper, the minimum distance distribution of irregular generalized
LDPC (GLDPC) code ensembles is investigated. Two classes of GLDPC code
ensembles are analyzed; in one case, the Tanner graph is regular from the
variable node perspective, and in the other case the Tanner graph is completely
unstructured and irregular. In particular, for the former ensemble class we
determine exactly which ensembles have minimum distance growing linearly with
the block length with probability approaching unity with increasing block
length. This work extends previous results concerning LDPC and regular GLDPC
codes to the case where a hybrid mixture of check node types is used.Comment: 5 pages, 1 figure. Submitted to the IEEE International Symposium on
Information Theory (ISIT) 201
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
Stability of Iterative Decoding of Multi-Edge Type Doubly-Generalized LDPC Codes Over the BEC
Using the EXIT chart approach, a necessary and sufficient condition is
developed for the local stability of iterative decoding of multi-edge type
(MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In
such code ensembles, the use of arbitrary linear block codes as component codes
is combined with the further design of local Tanner graph connectivity through
the use of multiple edge types. The stability condition for these code
ensembles is shown to be succinctly described in terms of the value of the
spectral radius of an appropriately defined polynomial matrix.Comment: 6 pages, 3 figures. Presented at Globecom 2011, Houston, T
On a Class of Doubly-Generalized LDPC Codes with Single Parity-Check Variable Nodes
A class of doubly-generalized low-density parity-check (D-GLDPC) codes, where
single parity-check (SPC) codes are used as variable nodes (VNs), is
investigated. An expression for the growth rate of the weight distribution of
any D-GLDPC ensemble with a uniform check node (CN) set is presented at first,
together with an analytical technique for its efficient evaluation. These tools
are then used for detailed analysis of a case study, namely, a rate-1/2 D-GLDPC
ensemble where all the CNs are (7,4) Hamming codes and all the VNs are length-7
SPC codes. It is illustrated how the VN representations can heavily affect the
code properties and how different VN representations can be combined within the
same graph to enhance some of the code parameters. The analysis is conducted
over the binary erasure channel. Interesting features of the new codes include
the capability of achieving a good compromise between waterfall and error floor
performance while preserving graphical regularity, and values of threshold
outperforming LDPC counterparts.Comment: 2009 IEEE Int. Symp. on Information Theory. 5 pages, 3 figure
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
A Probabilistic Peeling Decoder to Efficiently Analyze Generalized LDPC Codes Over the BEC
In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely, Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code and its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity, but the optimal fraction is strictly smaller than one. We then consider techniques to further reduce the gap to capacity by means of random puncturing, and the inclusion of a certain fraction of generalized variable nodes in the graph.This work was supported in part by the Spanish Ministerio de EconomÃa y Competitividad and the Agencia Española de Investigación under Grant TEC2016-78434-C3-3-R (AEI/FEDER, EU) and in part by the Comunidad de Madrid in Spain under Grant S2103/ICE-2845, Grant IND2017/TIC-7618, Grant IND2018/TIC-9649, and Grant Y2018/TCS-4705.
P. M. Olmos was further supported by the Spanish Ministerio de EconomÃa y Competitividad under Grant IJCI-2014-19150. T. Koch was further supported by the European Research Council (ERC) through the European Union’s
Horizon 2020 research and innovation programme under Grant 714161, by the 7th European Union Framework Programme under Grant 333680, and by the Spanish Ministerio de EconomÃa y Competitividad under Grant TEC2013-
41718-R and Grant RYC-2014-16332
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