LDPC Ensembles Research Articles

Design and Analysis of NB QC-LDPC Codes Over Small Alphabets

We propose a novel approach to optimization of irregular nonbinary (NB) quasi-cyclic (QC)-LDPC codes over small alphabets. In this approach, first, the base parity-check matrices are constructed by a simulated annealing method, and then these matrices are labeled by the field elements while maximizing the so- called generalized girth of the Tanner graph. In order to analyze the performance of the constructed irregular NB LDPC codes, a new ensemble of irregular NB LDPC codes over the extensions of the binary Galois field is introduced. A finite-length random coding bound on the error probability of the maximum-likelihood (ML) decoding over the binary phase shift keying (BPSK) input AWGN channel for the new code ensemble is derived. The frame error rate (FER) performance of the sum-product belief-propagation (BP) decoding of the constructed NB QC-LDPC block codes is compared to that of both the optimized binary QC-LDPC block codes in the 5G standard and the best known NB QC-LDPC codes as well as to the derived random coding bound on the ML decoding error probability. It is shown that the obtained bound predicts the behavior of BP decoding performance of practical NB QC-LDPC codes more accurately than the BP decoding thresholds do.

Design of Bilayer and Multi-Layer LDPC Ensembles From Individual Degree Distributions

A new approach for designing bilayer and multi-layer LDPC codes is proposed and studied in the asymptotic regime. The ensembles are defined through individual uni-variate degree distributions, one for each layer. We present a construction that: 1) enables low-complexity decoding for high-SNR channel instances, 2) provably approaches capacity for low-SNR instances, 3) scales linearly (in terms of design complexity) in the number of layers. For the setup where decoding the second layer is significantly more costly than the first layer, we propose an optimal-cost decoding schedule and study the trade-off between code rate and decoding cost.

Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

Regular spatially-coupled low-density parity-check ensembles have gained significant interest, since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this paper, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per code word and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability (PB) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random location in the code word and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability p, potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations.

Improving Windowed Decoding of SC LDPC Codes by Effective Decoding Termination, Message Reuse, and Amplification

In this paper, we address a number of weaknesses of the windowed decoding of spatially coupled low-density parity-check (SC LDPC) codes and propose three modifications that simultaneously improve its performance, complexity, and latency. An effective termination method of the windowed decoding and the reuse of edge messages of previous target symbols provide a good performance-latency tradeoff when compared with the conventional windowed decoder. Also, we propose a scheme that lowers the error floor, in which the amplified edge messages of the previous window are used in the present window. The proposed windowed decoding, consisting of the three schemes, provides a significant performance gain with smaller latency. The validity of the new windowed decoding is verified by the evaluation with codes from different SC LDPC ensembles.

Threshold Saturation for Nonbinary SC-LDPC Ensembles on Discrete-Input Memoryless Channels

In this letter, we investigate the threshold saturation effect of nonbinary spatially coupled low-density parity-check (SC-LDPC) ensembles defined over GF( $ {q}$ ) when the transmission takes place over discrete-input memoryless channels. To this end, we extend the duality rule for entropy established in the analysis of binary LDPC ensembles to the nonbinary case. This enables us to derive potential functions associated with log-likelihood ratio vector densities. Finally, we prove the existence of the threshold saturation effect following the approach introduced by Yedla et al. The analysis in this letter generalizes the existing work on nonbinary SC-LDPC ensembles and implies that the proof technique based on potential functions is also applicable to nonbinary sparse graphical codes over nonbinary discrete-input channels under belief propagation decoding.

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felström and Zigangirov, perform so well over this channel. We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in this ensemble have this property. The quantifier universal refers to the single ensemble/code that is good for all channels but we assume that the channel is known at the receiver. The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble. We conclude by discussing some interesting open problems.

Performance Analysis and Design of Two Edge-Type LDPC Codes for the BEC Wiretap Channel

We consider transmission over a wiretap channel where both the main channel and the wiretapper's channel are Binary Erasure Channels (BEC). We propose a code construction method using two edge type LDPC codes based on the coset encoding scheme. Using a standard LDPC ensemble with a given threshold over the BEC, we give a construction for a two edge type LDPC ensemble with the same threshold. If the given standard LDPC ensemble has degree two variable nodes, our construction gives rise to degree one variable nodes in the code used over the main channel. This results in zero threshold over the main channel. In order to circumvent this problem, we numerically optimize the degree distribution of the two edge type LDPC ensemble. We find that the resulting ensembles are able to perform close to the boundary of the rate-equivocation region of the wiretap channel. There are two performance criteria for a coding scheme used over a wiretap channel: reliability and secrecy. The reliability measure corresponds to the probability of decoding error for the intended receiver. This can be easily measured using density evolution recursion. However, it is more challenging to characterize secrecy, corresponding to the equivocation of the message for the wiretapper. M\'easson, Montanari, and Urbanke have shown how the equivocation can be measured for a broad range of standard LDPC ensembles for transmission over the BEC under the point-to-point setup. By generalizing the method of M\'easson, Montanari, and Urbanke to two edge type LDPC ensembles, we show how the equivocation for the wiretapper can be computed. We find that relatively simple constructions give very good secrecy performance and are close to the secrecy capacity. However finding explicit sequences of two edge type LDPC ensembles which achieve secrecy capacity is a more difficult problem. We pose it as an interesting open problem.

Reweighted LP Decoding for LDPC Codes

We introduce a novel algorithm for decoding binary linear codes by linear programming (LP). We build on the LP decoding algorithm of Feldman and introduce a postprocessing step that solves a second linear program that reweights the objective function based on the outcome of the original LP decoder output. Our analysis shows that for some LDPC ensembles we can improve the provable threshold guarantees compared to standard LP decoding. We also show significant empirical performance gains for the reweighted LP decoding algorithm with very small additional computational complexity.

Stability Outage Analysis for LDPC Codes

For a given channel instance and a fixed LDPC ensemble, a stability outage event is defined by the density evolution being unstable at the origin. The probability of such an event, referred to as Stability Outage Probability, is a useful lower bound for the word error rate. We formulate the stability outage probability for block fading channels and we introduce the stability diversity order. As a direct application, we show how the class of capacity-achieving ensembles on the erasure channel is bounded away from the outage limit on a block fading channel.

Some Results on MAP Decoding of Non-Binary LDPC Codes Over the BEC

In this paper, the transmission over the binary erasure channel (BEC) using non-binary LDPC (NBLDPC) codes is considered. The concept of peeling decoder and stopping sets is generalized to NBLDPC codes. Using these generalizations, a combinatorial characterization of decoding failures of NBLDPC codes is given, under assumption that the Belief Propagation (BP) decoder is used. Then, the residual ensemble of codes resulted by the BP decoder is defined and the design rate and the expectation of total number of codewords of the residual ensemble are computed. The decoding failure criterion combined with the density evolution analysis helps us to compute the asymptotic residual degree distribution for NBLDPC codes. Our approach to compute the residual degree distribution on the check node side is not efficient as it is based on enumeration of all the possible connections on the check node side which satisfy the decoding failure criterion. So, the computation of the asymptotic check node side residual degree distribution and further part of our analysis is performed for NBLDPC codes over GF2m with m = 2 . In order to show that asymptotically almost every code in such LDPC ensemble has a rate equal to the design rate, we generalize the argument of the Maxwell construction to NBLDPC codes, defined over GF22. It is also observed that, like in the binary setting, the Maxwell construction, relating the performance of MAP and BP decoding holds in this setting.

Convergence analysis of generalized serial message-passing schedules

Schedule is the order of passing messages between vertices of the bipartite graph defining an LDPC code in the process of decoding. Schedules affect the rate of decoding convergence. New efficient generalized serial schedules are described and analyzed, exhibiting significantly faster convergence compared to previously known schedules. For the proposed schedules, combinatorial and probabilistic analysis is presented, explaining the fast convergence observed in simulations. Using it, LDPC ensembles for which significantly better convergence rates can be achieved are identified. Specific code constructions from lifted graphs are further proposed, efficiently supporting the schedules. Examples based on regular LDPC codes are provided, in which the schedules achieve convergence speedup factors of up to 6 in comparison with the flooding schedule. Higher speedup factors are predicted by the analysis for irregular codes.

Slepian-Wolf Coding of Individual Sequences Based on Ensembles of Linear Functions

This paper clarifies the adequacy of the linear channel coding approach for Slepian-Wolf coding of individual sequences. A sufficient condition for ensembles of linear codes from which a universal Slepian-Wolf code can be constructed is given. Our result reveals that an ensemble of LDPC codes gives a universal code for Slepian-Wolf coding of individual sequences.

On the Asymptotic Weight and Stopping Set Distribution of Regular LDPC Ensembles

We estimate the variance of weight and stopping set distribution of regular LDPC ensembles. Using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular LDPC ensemble has its weight distribution and stopping set distribution close to respective ensemble averages. We are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average.

Decoding of low-density codes with parity-check matrices composed of permutation matrices in an erasure channel

A lower bound for the number of iteratively correctable erasures is given, with application to the ensemble of LDPC codes with parity-check matrices composed of permutation matrices [1]. We assume that the Zyablov-Pinsker iterative decoding algorithm [2] is used. Its complexity is O(N log N), where N is the block length.