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.
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 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.
In this paper we are concerned with the asymptotic analysis of nonbinary spatially-coupled low-density parity-check (SC-LDPC) ensembles defined over GL(2m) (the general linear group of degree m over GF(2)). Our purpose is to prove threshold saturation when the transmission takes place on the binary erasure channel (BEC). To this end, we establish the duality rule for entropy for nonbinary variable-node (VN) and check-node (CN) convolutional operators to accommodate the nonbinary density evolution (DE) analysis. Based on this, we construct the explicit forms of the potential functions for uncoupled and coupled DE recursions. In addition, we show that these functions exhibit similar monotonicity properties as those for binary LDPC and SC-LDPC ensembles over general binary memoryless symmetric (BMS) channels. This leads to the threshold saturation theorem and its converse for nonbinary SC-LDPC ensembles on the BEC, following the proof technique developed by S. Kumar et al.
We analyze the asymptotic performance of non-binary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group, when the transmission takes place over the binary erasure channel. We propose an efficient method to derive an upper bound to the maximum a posteriori probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP performance of regular LDPC codes improves with the alphabet size. We then consider nonbinary SC-LDPC codes. We show that the same threshold saturation effect experienced by binary SC-LDPC codes occurs for the nonbinary codes, hence we conjecture that the BP threshold for large termination length approaches the MAP threshold of the underlying regular ensemble.
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 has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian 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, binary asymmetric 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
In this paper, we present several theoretic results on density evolution (DE) for nonbinary spatially-coupled low-density parity-check (SC-LDPC) ensembles when the transmission takes place on the binary erasure channel (BEC). In specific, we establish the duality rule for entropy for the nonbinary variable-node (VN) and check-node (CN) operators in such a scenario. We define the partial order between densities and show that the VN and CN operators exhibit the property of partial order preservation. More importantly, we explicitly construct the potential functions for uncoupled and coupled DE recursions, the forms of which almost coincide with those for binary LDPC and SC-LDPC ensembles over general binary memoryless symmetric (BMS) channels. These theoretic findings greatly facilitate the proof of threshold saturation for nonbinary SC-LDPC ensembles on the BEC. Finally, we develop the threshold saturation theorem and its converse, following the lines established by S. Kumar et al.
LDPC codes over non-binary Galois fields were proposed for reliable transmission over arbitrary discrete memoryless channels and in particular for transmission in the bandwidth efficient regime. It was shown that some properties, originally derived for binary LDPC codes, extend to the appropriately defined non-binary ensemble. However, the extension of density evolution to this ensemble results in a prohibitive computational complexity. In this paper we analyze the performance by tracking the evolution of the Bhattacharya noise parameter associated with the decoding messages, and derive upper and lower bounds on its evolution. These bounds are then used to obtain upper and lower bounds on the asymptotic threshold required for reliable communication.