Extrinsic Message Research Articles

Design and analysis of stochastic 5G new radio LDPC decoder using adaptive sparse quantization kernel least mean square algorithm for optical satellite communications

AbstractA Stochastic Low‐Density Parity‐Check (LDPC) decoder is a type of 5G New Radio standard LDPC decoder that uses stochastic techniques to perform decoding. Stochastic LDPC decoding with 5G NR standard typically uses an iterative process, where messages exchanged among variable nodes (VN), check nodes multiple times. Stochastic LDPC decoders are often used in scenarios where the received signal is subject to varying levels of noise. They will provide improved error correction performance compared to traditional LDPC decoders, especially when dealing with channels with varying signal‐to‐noise ratios in 5G networks. Using the adaptive sparse quantization kernel least mean square algorithm (SLDPC‐ASQ‐KLMSA), this paper proposes an area‐efficient architecture design for a stochastic LDPC decoder. The LDPC code (2048, 1723) is taken from the LOGBASE‐T standard and used in this study. We examine the ASQ‐KLMSA connection effects. Starting with the VN. It makes checking node functioning easier and reduces inter‐connect complexity by capping extrinsic message length at 2 bits. Because of the simplified check node operation in ASQ‐KLMSA, the decoder nodes must exchange messages with a greater degree of accuracy. The 3–3 input grouping sub‐node of the degree‐6 VN was changed with an adder‐based 5–1 input grouping sub‐node for the (2048, 1723) code in order to get more accurate results when the check‐to‐variable messages aren't strong enough. A suggested decoder architecture was determined using a stochastic LDPC decoder developed for TSMC 65 nm process (2048, 1723). Bite error rate, throughput, mean square error, latency, power, and area usage are some of the metrics used to evaluate the effectiveness of the SLDPC‐ASQ‐KLMSA algorithm that has been suggested and implemented in Python. Thus, the proposed approach has attained 34.44%, and 38.39% low mean square error while compared with the existing methods such as higher‐performance stochastic LDPC decoder architecture designed through correlation analysis (HP‐SLDPC‐CA), Higher Throughput and Hardware Efficient Hybrid LDPC Decoder Utilizing Bit‐Serial Stochastic Updating(HLDPC‐BSSU), Flexible FPGA‐Based Stochastic Decoder for 5G LDPC codes (FPGA‐SD‐5G‐LDPC), respectively.

The Role of Fidelity in Goal-Oriented Semantic Communication: A Rate Distortion Approach

We study a variant of a robust description source coding framework, which is a relevant model for goal-oriented semantic information transmission, via its corresponding characterization. Considering two individual single-letter separable distortion constraints and input and output data acting as the intrinsic and extrinsic message, respectively, we first derive a lower bound on the optimal rates of the problem, as well as necessary and sufficient conditions for this bound to be tight. Subsequently, we prove a general result that provides in parametric form the optimal solution of the characterization of this problem. Capitalizing on these results, we examine the structure of the solution for one case study of general binary alphabets under Hamming distortions and solve in closed form a special case. We also solve another general binary alphabet case where a Hamming and an erasure distortion coexist, as a means to highlight the importance of selecting the type of the distortion constraint in goal-oriented semantic communication. Furthermore, we develop a goal-oriented Blahut-Arimoto (BA) algorithm, which can be used for the computation of any finite alphabet intrinsic or extrinsic message under individual distortion criteria. Finally, we revisit the problem for multidimensional independent and identically distributed (i.i.d.) jointly Gaussian processes with individual mean-square error (MSE) distortion constraints, providing new insights that have previously been overlooked. This work reveals the cardinal role of context-dependent fidelity criteria in goal-oriented semantic communication.

Framing Effects on Attitudes and Intentions Toward Shark Meat Consumption in Lima, Peru

Peru has one of the major shark fisheries in the world. Moreover, shark meat consumption is popular and the main commercially exploited species are considered threatened. Recent studies have found high mislabeling rates and high concentrations of methylmercury in shark meat. The purpose of this study is to explore the effectiveness of different framing messages in persuading fish consumers to avoid shark meat consumption and promote sustainable fish (“popular fish”) consumption. Specifically: To what extent do intrinsic and extrinsic risk messages differ in terms of deterring the intention to eat shark meat and promoting the consumption of popular fish, in the presence or absence of an efficacy message about mislabeling of shark meat? The three message frames were: intrinsic (sharks have mercury), extrinsic (sharks are threatened), and efficacy (sharks are mislabeled). The experiment had a factorial design of 2 × 2 × 2 (present vs. absent for each message). Participants (n = 285), surveyed through an online questionnaire, were assigned to one of eight experimental conditions. Participants then completed a survey regarding their intentions and attitudes toward shark meat and popular fish consumption. Attitudes and intentions toward consuming shark meat were discouraged in all experimental conditions. Moreover, the mislabeling message did not reduce participants' perceived response efficacy (how effective the recommendations for dealing with the problem are), but enhanced it. This study suggests that not all consumers' attitudes about sharks are negative or that a negative attitude does not affect shark consumption in an adverse way. More research targeting different audiences is needed to determine the optimal approach for promoting sustainable seafood that is also healthy for consumers.

High-Performance and Low-Complexity Decoding Algorithms for 5G Low-Density Parity-Check Codes

The development of the Fifth Generation (5G) New Radio (NR) provides several significant advantages when compared to the fourth generation (4G) Long Term Evolution (LTE) in mobile communications. Due to the outstanding characteristics of Low-Density Parity-Check (LDPC) codes such as high decoding performance, high throughput, low complexity, they have been accepted as the standard codes for the 5G NR. In this paper, we propose two LDPC decoding algorithms: Hybrid Offset Min-Sum (HOMS) and Variable Offset Min-Sum (VOMS), which are aimed at improving the error correction performance. The main idea of the HOMS/VOMS algorithm is to apply modified factors to both variable-nodes and check-nodes updated processing in order to compensate the extrinsic messages overestimation of the MS-based algorithms and increase the protection ability for degree-1 variable-nodes. The simulation results show that at the Bit-Error-Rate (BER) of 10-5, the proposed HOMS/VOMS algorithm achieves the decoding gain up to 0.2 dB compared to the Offset Min-Sum (OMS) algorithm, with a slight increase in decoding complexity.

Adults’ Preferences for Intrinsically versus Extrinsically Framed Health Messages Tailored According to Stages of Change: Effects on the Intention to Engage in Physical Activity

Physical activity rates have prevailed as a worldwide health concern. The WHO’s Global Action Plan on Physical Activity 2018–2030 articulated the need to test effective communication strategies that deliver information to enhance physical activity behaviors. The Comprehensive Messaging Strategy for Sustained Behavior Change (CMSSBC) is a promising approach that advocates tailoring messages to one’s stage of change and framing them intrinsically rather than extrinsically. This study examined adults’ decisions to read and engage with physical activity messages aligned with CMSSBC, considering the role of the message frame, stage of change of the message, stage of change of the participant, as well as the participants goal orientation. Participants (nmale = 91; nfemale = 103) aged 25 to 65 (x = 39.53; SD = 10.80) completed an online survey including; demographic questions, assessments of their physical activity goals and stage of change, decision to read eight messages, and a follow-up question providing the option to complete a behavior change task embedded within the messages they chose to read. Results indicated that participants were more likely to choose to read a message if it fit their goal orientation, and overall, they were significantly more likely to read intrinsic as opposed to extrinsic messages. Further, findings demonstrated that participants were more likely to complete the optional behavior change activity contained within the message if they had an intrinsic goal orientation. These findings demonstrate that physical activity information should be framed intrinsically and recognizes that one’s goal orientation may significantly influence how people respond to messages.

Refined Reliability Combining for Binary Message Passing Decoding of Product Codes

We propose a novel soft-aided iterative decoding algorithm for product codes (PCs). The proposed algorithm, named iterative bounded distance decoding with combined reliability (iBDD-CR), enhances the conventional iterative bounded distance decoding (iBDD) of PCs by exploiting some level of soft information. In particular, iBDD-CR can be seen as a modification of iBDD where the hard decisions of the row and column decoders are made based on a reliability estimate of the BDD outputs. The reliability estimates are derived using extrinsic message passing for generalized low-density-parity check (GLDPC) ensembles, which encompass PCs. We perform a density evolution analysis of iBDD-CR for transmission over the additive white Gaussian noise channel for the GLDPC ensemble. We consider both binary transmission and bit-interleaved coded modulation with quadrature amplitude modulation.We show that iBDD-CR achieves performance gains up to $0.51$ dB compared to iBDD with the same internal decoder data flow. This makes the algorithm an attractive solution for very high-throughput applications such as fiber-optic communications.

Lifting method for quasi-cyclic low-density parity-check codes based on approximated cycle extrinsic message degree in optical communication systems

Quasi-cyclic low-density parity-check (QC-LDPC) codes with low-complexity hardware implementation and excellent performance have become a hot research topic in many optical communication systems. To meet the different requirements of code lengths in optical communication systems, lifting methods which remove all the short cycles equally are usually used to construct long QC-LDPC codes of larger lengths by lifting short QC-LDPC codes. A lifting method is proposed, which considers both length and connectivity of short cycles. The proposed lifting method reduces the complexity of calculating the approximated cycle extrinsic message degree -related metrics and improves connectivity of the Tanner graph. The numerical results prove that, by employing our proposed lifting method, the lifted long QC-LDPC codes have better bit error rate performance compared with the lifted codes employing the traditional lifting method.

Improving Polar-Coded SCMA System by Information Coupling and Parity Check.

In this paper, the uplink information-coupled polar-coded sparse code multiple access (PC-SCMA) system is proposed. For this system, we first design the encoding method of systematic joint parity check and CRC-aided (PCCA) polar code. Using the systematic PCCA-polar code as base code, the partially information-coupled (PIC) polar code is constructed. Then, a joint iterative detection and successive cancellation list (SCL)-decoding receiver is proposed for the PC-SCMA system. For the receiver, the coupled polar decoder’s extrinsic messages are calculated by the Bayes rule and soft cancellation (SCAN) algorithm. Based on the extrinsic information transfer (EXIT) idea, the PIC PCCA-polar code is optimized. Simulation results demonstrate that the PIC PCCA-PC-SCMA system outperforms the other polar (or LDPC) coded SCMA systems at various code rates and channel configurations. Additionally, compared with an uncoupled PC-SCMA system with SCL decoder, the complexity of PIC PCCA-PC-SCMA is reduced at a high

Promoting Strength Training Among Baby Boomers: Message Framing Effects on Motivation and Behavior.

Regular participation in strength training significantly enhances older adults' physical and functional health, yet only 14% of adults over age 50 report engaging in strength training at least twice per week. Exercise messages about why and how older adults should engage in strength training could influence their perceptions of and participation in the behavior. This study was designed to promote strength training among adults aged 50-70 and to determine whether the message frame accompanying a brief workout would have an impact on participants' motivation, self-efficacy, and/or physical activity behavior. Participants (N=308) were randomly assigned to one of four message conditions: Easy Intrinsic, Hard Intrinsic, Easy Extrinsic, or Hard Extrinsic. Immediately after completing baseline measures of motivation and physical activity, participants received the recommended strength training workout and corresponding motivational message. They completed follow-up measures of motivation and physical activity 2weeks later. Controlled regulation was higher at follow-up among participants exposed to an extrinsic message relative to an intrinsic message (F(3,208)=3.08, p=.03). Strength training (Z=-5.95, p<.001) and moderate physical activity (Z=-3.23, p=.001) increased among participants not engaging in regular strength training at baseline. These results suggest message framing may have a short-term impact on Baby Boomers' physical activity motivation, such that emphasizing extrinsic goals (i.e., weight loss and appearance) may contribute to controlled motivation. However, repeated measurements are needed to determine the longer-term maintenance and influence of these effects.

LDPC-coded Probabilistic Shaping PAM4 Based on Many-to-One Mapping in WDM-PON

In this article, a 5 × 25-GBaud wavelength division multiplexing passive optical network (WDM-PON) is reported and experimentally demonstrated with each channel carrying a low-density parity check- (LDPC-) coded probabilistic shaping 4-ary pulse amplitude modulated signal (PS-PAM4). In such a splitter-based WDM-PON system, 8-point 3-bits/symbol PAM8 is mapped into 4-point PAM4 by adoption of a many-to-one mapping scheme, which enables improved receiver power sensitivity and increased spectral efficiency, compared with the uniformly-distributed PAM4. Different from the typical constant composition distribution matching method for PS, no shaping codes or PS overheads are needed in our PS-PAM4 signal which is to be encoded and decoded in a LDPC-based bit-interleaved coded modulation iterative decoding (BICM-ID) system for the compatibility with the forward error correction (FEC) applied in most of the optical transponders currently. In the BICM-ID system, the extrinsic messages are delivered circularly between the PS de-mapper and the LDPC decoder to deal with PS many-to-one mapping ambiguity. The experimental results strive to make the WDM-PON system better spectrum efficiency, as well as superior receiver power sensitivity and fiber nonlinear effect tolerance, thus maximizing the input power at the optical line terminal and economic efficiency of the system due to ampler loss budget provided for the legacy optical distribution networks.

Low-Complexity CRC Aided Joint Iterative Detection and SCL Decoding Receiver of Polar Coded SCMA System

As the fifth-generation (5G) wireless networks’ key technology, the joint design of SCMA and polar code is concerned by the future communication system. In this paper, a CRC aided joint iterative detection and successive cancellation list (SCL) decoding (CAJIDS) receiver is proposed for the polar coded SCMA (PC-SCMA) system. For the receiver, the SCL decoder’s extrinsic messages construction algorithm is designed by using Bayes rule and soft cancellation algorithm. Additionally, the distributed CRC aided polar (DCA-polar) code and variable list size are used to reduce the receiver’s decoding latency and complexity. Simulation results demonstrate that the CAJIDS receiver has better error rate performance than the joint iterative detection and decoding (JIDD) receiver. It also outperforms the LDPC coded SCMA (LDPC-SCMA) system. Specifically, when code length $N=256$ and code rate $R = 1/3$ , the CAJIDS outperforms the JIDD and LDPC-SCMA 1.2 dB and 1.9 dB over the Rayleigh fading channel, respectively. It also shows that, compared with the fixed-list-size receiver, with a similar error rate performance, the CAJIDS receiver has lower decoding latency and complexity.

On Orthogonal AMP in Coded Linear Vector Systems

Linear minimum mean square error (LMMSE) estimation based turbo detection has been extensively studied for coded linear systems since the seminal work of Wang and Poor (WP). The WP algorithm operates iteratively between a linear detector (LD) and a nonlinear detector (NLD): the LD suppresses the interference based on LMMSE filtering, and the NLD decodes the data by treating the output of the LD as an observation from an additive white Gaussian noise (AWGN) channel. In WP, the messages exchanged between LD and NLD are required to be extrinsic . For the NLD, the extrinsic message comes from the constraint imposed on feedforward error correction (FEC) codes. Therefore, WP does not work in an un-coded linear system. Recently, we proposed an orthogonal approximate message passing (OAMP) algorithm, which only requires the input/output error terms of LD and NLD to be orthogonal . We conjectured that for un-coded linear systems that involve certain large random matrices, the dynamics of OAMP can be accurately characterized by state evolution (SE). In this paper, we consider a coded linear system and develop an extrinsic message aided OAMP (EMA-OAMP) algorithm. Similar to the un-coded case, EMA-OAMP relaxes the requirements on output messages to be orthogonal instead of extrinsic. We derive an SE procedure to characterize the performance of OAMP in coded systems. We conjecture that this SE procedure is accurate, which is verified by simulation results. Under this conjecture, we show that EMA-OAMP can outperform WP under certain standard assumptions for iterative decoding. Extensive simulations results are provided to verify the advantages of OAMP in coded MIMO systems.

DFT-Spread DMT-WDM-PON Employing LDPC-Coded Probabilistic Shaping 16 QAM

A 5 × 100-Gb/s discrete-Fourier-transform spread (DFT-spread) discrete multi-tone wavelength division multiplexing passive optical network (DMT-WDM-PON) employing low-density parity-check (LDPC) coded probabilistic shaping 16-ary quadrature amplitude modulation (PS-16 QAM) is proposed and experimentally demonstrated. We first introduce PS-16 QAM mapping scheme with 16-point constellation reshaped from 32 symbols into splitter-based DMT-WDM-PON. The DMT-WDM-PON employing PS-16 QAM can achieve superior receiver power sensitivity (RPS) and enhanced spectral efficiency (SE) compared with the unshaped one. Compatible with the forward error correction (FEC) implemented in most of the optical transponders today, an optimized bit-interleaved coded modulation iterative decoding (BICM-ID) system based on LDPC coding is proposed to encode the PS-16 QAM DMT signal without any extra shaping codes or overheads. Then the labeling optimization with the assistance of extrinsic information transfer (EXIT) chart is investigated, in which the extrinsic messages are transferred between the PS demapper and the LDPC decoder for retrieving the PS-16 QAM DMT signal. Furthermore, the experimental results show that 1.5-dB RPS improvement and 2-dB fiber nonlinear effect tolerance enhancement are respectively realized over a 20-km standard single-mode fiber (SSMF) link by using joint applications of DFT-spread scheme and PS-16 QAM mapping in proposed DMT-WDM-PON system, which can thus provide abundant system loss budget for the legacy optical distribution networks (ODNs).

Optimization of QC-LDPC Codes by Edge Exchange Method Based on ACE

Long quasi-cyclic low-density parity check (QC-LDPC) codes have been chosen as one of the forward error correction (FEC) schemes in passive optical network (PON) systems benefiting from the excellent performance. About how to get the appropriate LDPC codes, some traditional methods which focus on obtaining large girth or flat approximated cycle extrinsic message degree (ACE) spectrums are proposed for the design of LDPC codes. These methods which may still make the existence of small stopping sets containing short cycles probably are not so useful for long QC-LDPC codes. Recently, the parallel vector message passing (PMP) algorithm oriented-to decreasing the number of short cycles is proved to be an effective method to optimize LDPC codes. Since ignoring short cycles with poor connectivity which can induce error floor, this method cannot guarantee its effectiveness for long QC-LDPC codes. In this letter, an edge exchange optimization method based on the global properties of ACE and girth called EEA method is proposed, which tries to reduce cycles with low ACE in the order of cycle length. The numerical results prove that our EEA method can bring the improvements of error floor up to two orders of magnitude to long QC-LDPC codes and obviously outperforms PMP algorithm.

Memory Compact High-Speed QC-LDPC Decoder Based on FPGA

In this paper, two compact memory strategies for partially parallel QC-LDPC decoder architecture are proposed. By compacting several adjacent rows hard decisions and extrinsic messages into one memory entry, which not only reduces the number of memory banks for hard decisions, but also facilitates multiple data accesses per clock cycle so as to increase the throughput of decoder. We demonstrate significant high speed and area efficient benefits of using the proposed techniques with an FPGA implementation of a CCSDS LDPC decoder on Xilinx XC5VLX330 device. The result shows that our new decoder can operate at a maximum frequency of 250 MHz after place and route, and achieve a throughput up to 2 Gb/s at 14 iterations.

An Improved Network-Coded Multiple Access for Power-Balanced Non-Orthogonal Multiple Access

A recently proposed network-coded multiple access (NCMA), consisting of XOR-based channel decoding and multi-user complete decoding (MUD-CD), can significantly improve throughput over conventional successive interference cancellation (SIC) decoding under near power-balanced channels. In this paper, we propose an improved NCMA (I-NCMA) for non-orthogonal multiple access (NOMA) in the 5G wireless communications, consisting of multiple users and a single base station (BS). At the transmitter side, the proposed scheme enables all users to simultaneously transmit messages to the BS in the same time and frequency. At the receiver side, we propose a joint-user channel decoding (JU-CD) to jointly decode all user messages by a single decoder. Then, the extrinsic messages from JU-CD can be jointly utilized by MUD-CD for decoding all user messages. Also, other than conventional NCMA that mainly deals with two users, I-NCMA adapts to the NOMA networks with an arbitrary number of users. Simulation results indicate that the proposed I-NCMA achieves a better decoding performance than both SIC and conventional NCMA over near power-balanced channels in the low-to-medium SNR regime, while retaining a performance close to conventional NCMA in the high SNR regime.

A New Multi-Edge Metric-Constrained PEG Algorithm for Designing Binary LDPC Code With Improved Cycle-Structure

To obtain a better cycle-structure is still a challenge for the low-density parity-check (LDPC) code design. This paper formulates two metrics firstly so that the progressive edge-growth (PEG) algorithm and the approximate cycle extrinsic message degree (ACE) constrained PEG algorithm are unified into one integrated algorithm, called the metric-constrained PEG algorithm (M-PEGA). Then, as an improvement for the M-PEGA, the multi-edge metric-constrained PEG algorithm (MM-PEGA) is proposed based on two new concepts, the multi-edge local girth and the edge-trials. The MM-PEGA with the edge-trials, say a positive integer $r$, is called the $r$-edge M-PEGA, which constructs each edge of the non-quasi-cyclic (non-QC) LDPC code graph through selecting a check node whose $r$-edge local girth is optimal. In addition, to design the QC-LDPC codes with any predefined valid design parameters, as well as to detect and even to avoid generating the undetectable cycles in the QC-LDPC codes designed by the QC-PEG algorithm, the multi-edge metric constrained QC-PEG algorithm (MM-QC-PEGA) is proposed lastly. It is verified by the simulation results that increasing the edge-trials of the MM-PEGA/MM-QC-PEGA is expected to have a positive effect on the cycle-structures and the error performances of the LDPC codes designed by the MM-PEGA/MM-QC-PEGA.

Optimized Construction of Protograph G-LDPC Codes by Modified EXIT Chart and MACE for New-Generation Wireless Communications

In this paper, a new systematic construction of optimized protograph generalized low-density parity-check codes with good decoding threshold and low error floor is proposed. First, a typical code graph is generated by combining an accumulate-repeat-accumulate seed protograph with a Tanner graph extension of a simple linear block node. Subsequently, it is analyzed and optimized theoretically, especially with the puncture mechanism, by a modified extrinsic information transfer chart and an asymptotic weight distribution. The generated graph is then extended to a base matrix by a copy-and-permute procedure, accompanied with the matrix split, which is optimized by a progressive edge growth for good randomness and girth property. Finally, the proposed code matrix is created by replacing “1” in the above base matrix with square circulant sub-matrices, the offsets of which are searched by a quasi-cyclic (QC)-oriented modified approximate cycle extrinsic message degree algorithm to improve the cycle relationship, especially for the compound cycles. Simulation results show that the codes exhibit excellent performance in both error floor and waterfall region on an additive white Gaussian noise channel. Moreover, they are also characterized by faster encoding due to the QC structure as well as lower decoding complexity and less latency, which make them a natural fit for new-generation power constrained wireless communications.

Exploiting Cyclic Prefix for Turbo-OFDM Receiver Design

In an orthogonal frequency division multiplexing (OFDM) system, the cyclic prefix (CP) has traditionally been not useful for data detection at the receiver. This paper presents a low-complexity turbo-OFDM receiver to exploit all of the extra information offered by the CP to improve the data detection and channel estimation performance. In the receiver, two iterative processes, namely, the outer iteration based on the turbo equalization principle and the inner iteration based on the expectation–maximization algorithm, are proposed. The CP observation, which contains extra information about data and channel, is represented by a Forney-style factor graph and incorporated into the inner iteration for joint channel estimation and data detection, using a Gaussian message passing process. The extrinsic messages of the data enhanced by exploiting the CP are then exchanged in the outer iteration for joint channel estimation, data detection, and decoding to further improve a system performance. To effectively implement the receiver, several approximations are introduced to reduce the computational complexity to be logarithmic in the length of the OFDM symbol $N$ , i.e., $\mathcal O(NlogN)$ per OFDM symbol. Extensive simulation results show that the bit error rate performance of the proposed low-complexity turbo-OFDM receiver is superior to that of existing turbo-OFDM receivers with exploiting the channel information in the CP only or without exploiting the CP, as well as conventional OFDM receivers, in particular for time-varying channels.

Thresholds of Absorbing Sets in Low-Density Parity-Check Codes

The error floor phenomenon in many low-density parity-check (LDPC) codes is caused by combinatorial objects in their Tanner graph, known as absorbing sets . In this paper, we highlight a threshold behavior for the min-sum decoding algorithm in the graph of an absorbing set with fixed-point representation of messages. For an absorbing set of interest in a binary LDPC code we can compute the threshold , a novel real-valued parameter that is closely related to its harmfulness. We show that absorbing sets with negative thresholds cannot trap the decoder if the dynamic range of the extrinsic messages is large enough. We also prove that, in regular LDPC codes, absorbing sets with negative thresholds exist if the variable node degree is odd. The examples presented in this paper show that odd-column-weight LDPC codes can have many absorbing sets with negative thresholds, but that these absorbing sets do not trap a well-designed decoder. Simulations show a good agreement between the results of the analysis presented in this paper and the performance of practical decoders with fixed-point messages.