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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Detailedly Represented Irregular Low-Density Parity-Check Codes ∗

Kenta Kasai, Tomoharu Shibuya, Kohichi Sakaniwa

Published Oct 1, 2003• 26 citations
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Abstract

Richardson and Urbanke developed a powerful method density evolution which determines, for various channels, the capacity of irregular low-density parity-check code ensembles. We develop generalized density evolution for minutely represented ensembles and show it includes conventional representation as a special case. Furthermore, we present an example of code ensembles used over binary erasure channel and binary input additive white Gaussian noise channel which have better thresholds than highly optimized ensembles with conventional representation. key words: density evolution, irregular low-density parity-check codes, low-density parity-check codes, belief propagation

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IEEE International Symposium on Information Theory (ISIT)•Nov 13, 2002•5 citations
Richardson and Urbanke developed a powerful method density evolution which determines, for various channels, the capacity of irregular low-density parity-check code ensembles. We develop generalized density evolution for minutely represented ensembles and show it includes conventional representation as a special case. Furthermore, we present an example of code ensembles used over binary erasure channel and binary input additive white Gaussian noise channel which have better thresholds than highly optimized ensembles with conventional representation. key words: density evolution, irregular low-density parity-check codes, low-density parity-check codes, belief propagation
This paper extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF(q) . We first generalize the definition of channel symmetry for nonbinary inputs to include q-ary phase-shift keying (PSK) modulated channels for prime q and binary-modulated channels for q that is a power of 2. For the well-defined q-ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by (q-1) quantities. Assuming uniform edge weights, we further show that the density of messages computed by the check node decoder (CND) is fully defined by a single number. We then present the approximate density evolution for regular and irregular LDPC codes, and show that the (q-1) -dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q=2p. Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF(3) and GF(4). The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF(q) codes for high-order modulation by using the idea of a channel adapter. We find that codes designed in this fashion outperform those optimized specifically for the binary additive white Gaussian noise (AWGN) channel for a short codewords and a spectral efficiency of 2 bits per channel use (b/cu).
This work presents an ensemble of irregular low-density parity check (LDPC) codes based on an ensemble of bipartite graphs, which formulated the average weight distribution. An analysis of the error performance of LDPC codes over a binary erasure channel together with an iterative decoding algorithm based on belief propagation has been clarified that a notion of stopping sets and their distribution with the asymptotic expression are derived. An irregular LDPC code ensemble that exhibits better performance in the sense of threshold is obtained by density evolution and explicitly formulates weight and stopping set distributions are defined.
Abstract — Density evolution is one of the most powerful analytical tools for low-density parity-check (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for non-symmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. z-channels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels. Index Terms — Low-density parity-check (LDPC) codes, density evolution, sum-product algorithm, asymmetric channels, z-channels, rank of random matrices. I.
We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.