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arXiv:0801.2233 · IEEE International Symposium on Information Theory (ISIT)

Analysis of Non-binary Hybrid LDPC Codes

Lucile Sassatelli, David Declercq

Published Jun 1, 2007• 10 citations
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Abstract

This paper is eligible for the student paper award. In this paper, we analyse asymptotically a new class of LDPC codes called non-binary hybrid LDPC codes, which has been recently introduced in L. Sassatelli and D. Declerq [2006]. We use density evolution techniques to derive a stability condition for hybrid LDPC codes, and prove their threshold behavior. We study this stability condition to conclude on asymptotic advantages of hybrid LDPC codes compared to their non-hybrid counterparts.

DOI
10.1109/isit.2007.4557556

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This paper is eligible for the student paper award. In this paper, we analyse asymptotically a new class of LDPC codes called non-binary hybrid LDPC codes, which has been recently introduced in L. Sassatelli and D. Declerq [2006]. We use density evolution techniques to derive a stability condition for hybrid LDPC codes, and prove their threshold behavior. We study this stability condition to conclude on asymptotic advantages of hybrid LDPC codes compared to their non-hybrid counterparts.
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
The density evolution equations for non-binary low-density parity-check (LDPC) ensembles when transmission takes place over the binary erasure channel are derived. Ensembles defined with respect to the general linear group over the binary field are introduced. For these ensembles the density evolution equations can be written compactly. The density evolution for the general linear group helps in understanding the density evolution for codes defined with respect to finite fields. Thresholds are computed for different alphabet sizes for various LDPC ensembles. Surprisingly, the threshold is not a monotonic function of the alphabet size. The stability condition for non-binary LDPC ensembles over any binary memoryless symmetric channel is stated. Upper bounds on the MAP thresholds for various non-binary ensembles based on EXIT curves and the area theorem are given.
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.
In this paper we present a thorough analysis of non binary LDPC codes over the binary erasure channel. First, the decoding of non binary LDPC codes is investigated. The proposed algorithm performs ldquoon-the flyrdquo decoding, i.e. it starts decoding as soon as the first symbols are received, which generalizes the erasure decoding of binary LDPC codes. Next, we evaluate the asymptotical performance of ensembles of non binary LDPC codes, by using the density evolution method. Density evolution equations are derived by taking into consideration both the irregularity of the bipartite graph and the probability distribution of the graph edge labels. Finally, infinite-length performance of some ensembles of non binary LDPC codes for different edge label distributions are shown.