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International Conference on Software, Telecommunications and Computer Networks (SoftCOM)

Low Complexity Bounds on a Class of Irregular LDPC Belief-Propagation Decoding Thresholds

F. Vatta, Alessandro Soranzo, Massimiliano Comisso, Giulia Buttazzoni, Fulvio Babich

Published Sep 17, 2020• 1 citation
Publisher

Abstract

This paper investigates about the usefulness of a recently published low complexity upper bound on beliefpropagation decoding thresholds for a class of irregular lowdensity parity-check (LDPC) codes. In particular, the class considered is characterized by variable node degree distributions λ(x) of minimum degree ${i_1}$ >2: being, in this case, $\lambda^{\prime}$(0)=$\lambda_{2}=0$, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires $\lambda_{2}\neq0$. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.

DOI
10.23919/softcom50211.2020.9238285

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This paper investigates about the usefulness of a recently published low complexity upper bound on beliefpropagation decoding thresholds for a class of irregular lowdensity parity-check (LDPC) codes. In particular, the class considered is characterized by variable node degree distributions λ(x) of minimum degree i1{i_1}i1​ >2: being, in this case, λ′\lambda^{\prime}λ′(0)=λ2=0\lambda_{2}=0λ2​=0, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires λ2≠0\lambda_{2}\neq0λ2​=0. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.
This paper investigates the performance of a class of irregular low-density parity-check (LDPC) codes through a recently published low complexity upper bound on their beliefpropagation decoding thresholds. Moreover, their performance analysis is carried out through a recently published algorithmic method, presented in Babich et al. 2017 paper. In particular, the class considered is characterized by variable node degree distributions λ(x)\lambda(x)λ(x) of minimum degree i1>2i_1 \gt 2i1​>2: being, in this case, λ′(0)=λ2=0\lambda^{'} (0)=\lambda_2=0λ′(0)=λ2​=0, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires λ2≠0\lambda_2 \neq 0λ2​=0. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.
This paper investigates about the usefulness of some recently published low complexity upper bounds on belief-propagation decoding thresholds for irregular low-density parity-check (LDPC) codes to analyze their performance on memoryless binary input - additive white Gaussian noise (BI-AWGN) channels, with sum-product decoding. Irregular LDPC codes are known to perform better than regular ones, and to exhibit, like them, the so called “threshold phenomenon”, being the threshold defined as the maximum noise level such that an arbitrarily small bit-error probability can be achieved as the block length tends to infinity. We use a simplified analysis of the belief-propagation decoding algorithm, i.e., consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes on BI-AWGN channels.
This paper presents the design of block lengths for irregular low-density parity-check (LDPC) codes based on the maximum variable degree d v,max . To design a block length, the performance degradation of belief-propagation (BP) decoding performance from upper bounds on the maximum likelihood (ML) decoding performance is used as an important factor. Since for large block lengths, the performance of irregular LDPC codes is very close to the Shannon limit, we focus on moderate block lengths (5×10²≤N≤4×10³). Given degree distributions, the purpose of our paper is to find proper block lengths based on the maximum variable degree d v,max . We also present some simulation results which show how a block length can be optimized.
Since irregular low‐density parity‐check (LDPC) codes are known to perform better than regular ones, and to exhibit, like them, the so‐called ‘threshold phenomenon’, this Letter investigates a low‐complexity upper bound on belief‐propagation decoding thresholds for this class of codes on memoryless binary input additive white Gaussian noise channels, with sum‐product decoding. A simplified analysis of the belief‐propagation decoding algorithm is used, i.e. consider a Gaussian approximation for message densities under density evolution, and a simple algorithmic method, defined recently, to estimate the decoding thresholds for regular and irregular LDPC codes.
Low-density parity-check (LDPC) codes characterized by minimum Hamming distances proportional to block sizes have been demonstrated. Like the codes mentioned in the immediately preceding article, the present codes are error-correcting codes suitable for use in a variety of wireless data-communication systems that include noisy channels. The previously mentioned codes have low decoding thresholds and reasonably low error floors. However, the minimum Hamming distances of those codes do not grow linearly with code-block sizes. Codes that have this minimum-distance property exhibit very low error floors. Examples of such codes include regular LDPC codes with variable degrees of at least 3. Unfortunately, the decoding thresholds of regular LDPC codes are high. Hence, there is a need for LDPC codes characterized by both low decoding thresholds and, in order to obtain acceptably low error floors, minimum Hamming distances that are proportional to code-block sizes. The present codes were developed to satisfy this need. The minimum Hamming distances of the present codes have been shown, through consideration of ensemble-average weight enumerators, to be proportional to code block sizes. As in the cases of irregular ensembles, the properties of these codes are sensitive to the proportion of degree-2 variable nodes. A code having too few such nodes tends to have an iterative decoding threshold that is far from the capacity threshold. A code having too many such nodes tends not to exhibit a minimum distance that is proportional to block size. Results of computational simulations have shown that the decoding thresholds of codes of the present type are lower than those of regular LDPC codes. Included in the simulations were a few examples from a family of codes characterized by rates ranging from low to high and by thresholds that adhere closely to their respective channel capacity thresholds; the simulation results from these examples showed that the codes in question have low error floors as well as low decoding thresholds. As an example, the illustration shows the protograph (which represents the blueprint for overall construction) of one proposed code family for code rates greater than or equal to 1.2. Any size LDPC code can be obtained by copying the protograph structure N times, then permuting the edges. The illustration also provides Field Programmable Gate Array (FPGA) hardware performance simulations for this code family. In addition, the illustration provides minimum signal-to-noise ratios (Eb/No) in decibels (decoding thresholds) to achieve zero error rates as the code block size goes to infinity for various code rates. In comparison with the codes mentioned in the preceding article, these codes have slightly higher decoding thresholds.
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. A tightened union-type upper bound on the ML decoding error probability based on the precise coefficients of the average weight spectrum is presented. For LDPC codes from the Gallager ensemble and the Richardson-Urbanke ensemble, new upper bounds on the ML decoding performance based on computing the rank of submatrices of the code parity-check matrix are derived. A new lower bound on the ML decoding threshold followed from the latter error probability bound is obtained. An improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well. A new low-complexity near-ML decoding algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its performance is compared to the simulated belief propagation and ML decoding performance and simulated performance of the best known improved iterative decoding techniques, as well as, with the derived upper bounds on the ML decoding performance and with decoding thresholds obtained by the density evolution technique.