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Quelques Resultats sur les Codes LDPC Hybrides

Lucile Sassatelli, David Declercq

Published Jan 1, 2007• 0 citations
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Abstract

In this paper, some results on a very general class of LDPC codes are gathered. The hybrid LDPC codeword symbols belong to different order finite sets. That is why we can jointly control the couple (edge density,symbol size). Thanks to the stability condition given in this paper, we identify an advantage of hybrid LDPC codes over classical binary and non-binary LDPC codes. Information content evolution is analysed using a Gaussian approximation, in order to efficiently design hybrid LDPC codes. Some finite length constraints can be included in the asymptotic optimization, leading to good trade-off between convergence and error floor. In particular, for code rate 1/6, the optimized hybrid LDPC code is better than the best known codes.

Related papers

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In this paper, some results on a very general class of LDPC codes are gathered. The hybrid LDPC codeword symbols belong to different order finite sets. That is why we can jointly control the couple (edge density,symbol size). Thanks to the stability condition given in this paper, we identify an advantage of hybrid LDPC codes over classical binary and non-binary LDPC codes. Information content evolution is analysed using a Gaussian approximation, in order to efficiently design hybrid LDPC codes. Some finite length constraints can be included in the asymptotic optimization, leading to good trade-off between convergence and error floor. In particular, for code rate 1/6, the optimized hybrid LDPC code is better than the best known codes.
In this paper, a new class of low-density parity-check (LDPC) codes, named hybrid LDPC codes, is introduced. Hybrid LDPC codes are characterized by an irregular connectivity profile and heterogeneous orders of the symbols in the codeword. It is shown in particular that the class of hybrid LDPC codes can be asymptotically characterized and optimized using density evolution (DE) framework, and a technique to maximize the minimum distance of the code is presented. Numerical assessment of hybrid LDPC code performances is provided, by comparing them to protograph-based and multiedge-type (MET) LDPC codes. Hybrid LDPC codes are shown to allow to achieve an interesting tradeoff between good error-floor performance and good waterfall region with nonbinary coding techniques.
In this paper, we propose to study and optimize a very general class of LDPC codes whose variable nodes belong to finite sets with different orders. We named this class of codes hybrid LDPC codes. Although efficient optimization techniques exist for binary LDPC codes and more recently for non-binary LDFC codes, they both exhibit drawbacks due to different reasons. Our goal is to capitalize on the advantages of both families by building codes with binary (or small finite set order) and non-binary parts of their factor graph representation. The class of hybrid LDPC codes is obviously larger than existing types of codes, which gives more degrees of freedom to find good codes where the existing codes show their limits. We gave two examples where hybrid LDPC codes show their interest
In this paper, we use the class of non-binary hybrid LDPC codes to design very efficient low rate codes. To this end, we consider both asymptotic and finite length designs. First, we present an asymptotic analysis to design hybrid LDPC codes distributions and we explicit the cases when it is useful to use it. Then, by optimizing the algebraic properties of some topologies related to the Tanner graph, we design finite length codes. These codes show significant performance improvements both in the waterfall and in the error floor regions, in comparison with existing state-of-the-art low rate coding schemes, like turbo Hadamard or parallel concatenated Zigzag Hadamard codes, and with no increase of complexity. In particular, the minimum distance of low rate hybrid LDPC codes is by far greater than the ones of previously mentioned codes.
Time-invariant hybrid (HTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution and finite-length simulation to analyze the performance and the convergence properties of these algorithms. Tight upper bounds on the threshold of MB HTIalgorithms are derived, and it is proven that the asymptotic error probability for these algorithms tends to zero at least exponentially with the number of iterations. We devise optimal MB HTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HTIalgorithms are more robust, and can better cope with unknown channel conditions, and thus can be practically more attractive
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).