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arXiv:0810.4404 · International Symposium on Applied Sciences in Biomedical and Communication Technologies

Non binary LDPC codes over the binary erasure channel: Density evolution analysis

Valentin Savin

Published Oct 1, 2008• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/isabel.2008.4712621

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Allerton Conference on Communication, Control, and Computing•Jan 1, 2007•1 citation
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.
We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. The bit error probability for infinite block length is known by density evolution and it is well known that a difference between the bit error probability at finite iteration number for finite block length nnn and for infinite block length is asymptotically α/nα/nα/n, where ααα is a specific constant depending on the degree distribution, the iteration number and the erasure probability. Our main result is to derive an efficient algorithm for calculating ααα for regular ensembles. The approximation using ααα is accurate for (2,r)(2,r)(2,r)-regular ensembles even in small block length.
We generalize the results about how to compute iterative decoding thresholds over the binary erasure channel of non-binary LDPC code ensembles of [16] to non-binary TLDPC codes [2], [3]. We show in this case how density evolution can be performed in order to calculate iterative decoding thresholds and find several families with a very simple regular structure and thresholds close to the Shannon limit. To check the performances of these codes over other channels we have tested one of the simplest codes over F4which has rate 1/2 on the Gaussian channel. For the (binary) length 1008 for instance, without any optimization on the permutation structure of the code, it matches the performances of the best binary codes of the same length up to the word-error rate 10-3. We also notice that all LDPC codes (binary or not) having at least two symbols of degree 2 per parity-check equation can be represented as a special kind of TLDPC codes. We show that this representation and the associated decoding algorithm leads in the case of cycle codes to a significant reduction of the number of iterations which are needed for iterative decoding.
In the first part of this thesis we are interested in the asymptotic performance analysis of Non-Binary Low-Density Parity-Check (NBLDPC) codes over the Binary Erasure Channel (BEC) decoded via the suboptimal Belief Propagation (BP) decoder as well as the optimal Maximum-A-Posteriori (MAP) decoder. Previously, NBLDPC codes defined with respect to a finite field have been studied in the literature. Unfortunately, for these codes the asymptotic analysis of the BP decoder in terms of density evolution is cumbersome since the involved densities "live" in a high dimensional space. To alleviate this problem, we introduce ensembles defined with respect to the general linear group. For these ensembles the density evolution equations can be written in a compact form. We compute thresholds for different alphabet sizes for various NBLDPC ensembles. Surprisingly, the threshold is not a monotonic function of the alphabet size. We also conjecture the stability condition for NBLDPC ensembles defined with respect to finite fields over any binary memoryless symmetric channel. We then consider the performance of NBLDPC codes under MAP decoding when transmission takes place over the BEC. Towards this goal, we generalize the concepts of the peeling decoder and stopping sets to NBLDPC codes. Using these concepts, we give a combinatorial characterization of decoding failures for NBLDPC codes decoded via the BP decoder. We use the decoding failure criterion and the density evolution analysis to compute the asymptotic residual degree distribution for NBLDPC codes. The rate of this residual ensemble gives the conditional entropy of the NBLDPC code. To compute the rate of the residual ensemble we generalize the criterion of [1] to the non-binary setting, which, when satisfied, guarantees that almost every code in the ensemble has a rate equal to the design rate. We observe that the Maxwell construction of [1], relating the performance of MAP and BP decoding via the Extended BP GEXIT (EBP GEXIT [2]) function, holds in the setting of NBLDPC codes. The weight and stopping set distribution of a code are related to its performance under the MAP and the BP decoder respectively. We first concentrate on the weight distribution of regular NBLDPC ensembles. We derive the average weight distribution of regular NBLDPC ensembles. Using this, we derive the average of equivalent binary weight distribution of regular NBLDPC ensembles. We show that as the alphabet size becomes larger the equivalent binary weight distribution converges to the weight distribution of Gallager's random parity check ensemble. The average weight distribution of binary LDPC ensembles has been extensively studied in the literature. Much less is known about the probability distribution of this quantity. In particular, the question of the concentration of the weight distribution around the ensemble average has not been addressed. We compute the variance of the weight distribution of binary regular LDPC ensembles. Using this and the second moment method we obtain bounds on the probability that a randomly chosen code has its weight distribution close to the ensemble average. We show that a large fraction of the total number of codes have their weight distribution close to the average. We apply the same technique for the stopping set distribution of binary regular LDPC ensembles. Again we show that a large fraction of total number of codes have their stopping set distribution close to the ensemble average. In the last part of this thesis we address the question of the existence of the EBP GEXIT function. The EBP GEXIT function plays a fundamental role in the asymptotic analysis of sparse graph codes. For transmission over the BEC using binary LDPC codes, the analytic properties of the EBP GEXIT function are relatively simple and well understood. The case of general BMS channel is much harder and even the existence of the curve is not known in general. We introduce some tools from non-linear analysis which can be useful to prove the existence of EXIT like curves in some cases. The main tool is the Krasnoselskii-Rabinowitz (KR) bifurcation theorem.
This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.