Decoding Threshold Research Articles

What role, if any, should phonics play in a middle school or high school? The answer may surprise you

AbstractResearch shows little benefit from phonics instruction in Grades 2 through 12. However, more recent studies show that students who fall below a decoding threshold fail to benefit from other kinds of reading instruction. This exploration of the evidence suggests that these students are likely to need support in the reading and spelling of multisyllabic words and words with common morphological elements. Explicit instruction with a focus on the decoding, spelling, and meaning of such words would make a lot of sense.

Тізбекті кодтардың көп шекті декодтау параметрлерін оңтайландыру алгоритмін әзірлеу

The most known and practically used coding/decoding schemes are investigated. The results of the analysis of algorithmic complexity and simulation of the operation of coding schemes for Gaussian channels show that only multi-threshold decoders of convolutional codes outperform all others for all considered block sizes and coding rates. The multithreshold decoder (MTD) is a development of the simplest Messian threshold decoder with an additional difference register added. This decoder allows decoding block and convolutional self-orthogonal codes. The operating principles of convolutional encoders are considered in detail, the characteristics of multithreshold decoders and the Viterbi decoder for convolutional codes are compared. The best self-orthogonal codes for MTD are chosen, the efficiency of MTD in channels with additive white Gaussian noise is studied. An algorithm for increasing the performance of multithreshold decoding (MTD) based on the optimization of the MTD parameters of symbolic self-orthogonal codes is proposed.

Convolutional-Neural-Network-Based Hexagonal Quantum Error Correction Decoder

Topological quantum error-correcting codes are an important tool for realizing fault-tolerant quantum computers. Heavy hexagonal coding is a new class of quantum error-correcting coding that assigns physical and auxiliary qubits to the vertices and edges of a low-degree graph. The layout of heavy hexagonal codes is particularly suitable for superconducting qubit architectures to reduce frequency conflicts and crosstalk. Although various topological code decoders have been proposed, constructing the optimal decoder remains challenging. Machine learning is an effective decoding scheme for topological codes, and in this paper, we propose a machine learning heavy hexagonal decoder based on a convolutional neural network (CNN) to obtain the decoding threshold. We test our method on heavy hexagonal codes with code distance of three, five, and seven, and increase it to five, seven, and nine by optimizing the RestNet network architecture. Our results show that the decoder thresholding accuracies are about 0.57% and 0.65%, respectively, which are about 25% higher than the conventional decoding scheme under the depolarizing noise model. The proposed decoding architecture is also applicable to other topological code families.

Improving the Thresholds of Generalized LDPC Codes With Convolutional Code Constraints

CC-GLPDC codes are a class of generalized low-density parity-check (GLDPC) codes where the constraint nodes (CNs) represent convolutional codes. This allows for efficient decoding in the trellis with the forward-backward algorithm, and the strength of the component codes easily can be controlled by the encoder memory without changing the graph structure. In this letter, we extend the class of CC-GLDPC codes by introducing different types of irregularity at the CNs and investigating their effect on the BP and MAP decoding thresholds for the binary erasure channel (BEC). For the considered class of codes, an exhaustive grid search is performed to find the BP-optimized and MAP-optimized ensembles and compare their thresholds with the regular ensemble of the same design rate. The results show that irregularity can significantly improve the BP thresholds, whereas the thresholds of the MAP-optimized ensembles are only slightly different from the regular ensembles. Simulation results for the AWGN channel are presented as well and compared to the corresponding thresholds.

Rate-Compatible Protograph LDPC Codes for Source Coding in Joint Source—Channel Coded Modulation Systems

Using the source residual redundancy in a joint source and channel coded modulation (JSCCM) system to achieve a shaping gain has proven to be a good solution for probabilistic amplitude shaping. In a JSCCM system, rate-compatible source codes are essential for adapting to dynamically changing source probabilities. However, conventional rate-compatible source codes are not appropriate for JSCCM systems, since they may yield a loss in the shaping gain. Moreover, a code design that depends solely on the source decoding threshold appears to be ill suited, thus complicating the search for good source codes. In this paper, we propose a rate-compatible family of source codes for JSCCM systems based on an achievable system rate analysis and the source protograph extrinsic information transfer (PEXIT) algorithm. The proposed codes not only have good source decoding thresholds but also obtain shaping gains over the whole considered range of source probabilities. Numerical results show that the proposed rate-compatible family of source codes can significantly improve the bit-error-rate (BER) performance of the JSCCM system.

Operating Characteristic Curves of Optical Packet-Switching Using Optical Code-Division Multiplexing for Label Switching

In this paper, the operating characteristic curves (OCCs) of optical code-division multiplexing (OCDM) technology for label switching of an optical packet-switching (OPS) network was evaluated. A node structure for processing the packets, with spectral-amplitude-coding (SAC) labels, considering a balanced detector and an optical switch, was developed and modeled. The effects of decoding noises on the performance of both M-sequence and stuffed quadratic congruence (SQC) labeling systems were addressed. Hypothesis testing was applied to the decoder to investigate the results of label recognition. The null and alternative hypotheses were, respectively, defined as a decoder receiving the matching and mismatching labels. Due to the noise effects, the decoder output may not reflect the label status correctly. Type I error occurs when the null hypothesis is true while accepting the alternative one. Type II error occurs when the alternative hypothesis is true while accepting the null one. Analytic equations of both errors were given, considering a desired packet that was missed and an undesired packet shown in a switched path. The trade-off between these two errors, regarding the decoder threshold, was demonstrated in operating characteristic curves (OCCs). A better OCC could be found when a packet had more labeled payload bits, or when the utilized label code had a lower auto-to-cross-correlation ratio.

Construction of protograph LDPC codes based on the convolution neural network

This paper presents an intelligent protograph construction algorithm. Protograph LDPC codes have shown excellent error correction performance and play an important role in wireless communications. Random search or manual construction are often used to obtain a good protograph, but the efficiency is not high enough and many experience and skills are needed. In this paper, a fast searching algorithm is proposed using the convolution neural network to predict the iterative decoding thresholds of protograph LDPC codes effectively. A special input data transformation rule is applied to provide stronger generalization ability. The proposed algorithm converges faster than other algorithms. The iterative decoding threshold of the constructed proto-graph surpasses greedy algorithm and random search by about 0.53 dB and 0.93 dB respectively under 100 times of density evolution. Simulation results show that quasi-cyclic LDPC (QC-LDPC) codes constructed from the proposed algorithm have competitive performance compared to other papers.

Optimization of Protograph LDPC Codes via Surrogate Channel for Unequal Power Allocation

This paper investigates the possibilities of optimizing existing protograph low-density parity-check (PLDPC) codes and designing optimal PLDPC codes on both AWGN and Rayleigh channels under binary modulation with unequal power allocation (UPA) for nonuniform sources. While the binary UPA modulation achieves a power gain, it also results in unequal error protection on the coded bits in systematic codewords during transmission. For irregular codes, it is possible to best protect the information bits by optimizing the classification of variable nodes (VNs). Those VNs represent the coded bits in protograph that allows exploiting jointly the degree irregularity and the power differentiation. An algorithm for UPA-based protograph extrinsic information transfer (UPA-PEXIT) analysis on the Rayleigh channel is proposed with the concept of surrogate channel which circumvents the channel asymmetry problem with UPA. With the RJA code as example, it is shown that classification optimization does improve the decoding threshold when the source entropy is small. To design optimal codes, a UPA-based differential evolution (UPA-DE) algorithm is proposed in conjunction with the UPA-PEXIT analysis. A family of optimized codes are illustrated for three typical nonuniform source distributions and compared with the RJA and NND codes by simulations in error rate performance.

Spectrum-efficient user association in caching enabled dense small cell network

In this paper, we reveal the fundamental limitation of network densification on the performance of caching enabled small cell network (CSCN) under two typical user association rules, namely, content- and distance-based rules. It indicates that immoderately caching content would significantly change the interference distribution in CSCN, which may degrade the network area spectral efficiency (ASE). Meanwhile, it is shown that content-based rule outperforms the distance-based rule in terms of network ASE only when small cell base stations (BSs) are sparsely deployed with low decoding thresholds. Moreover, it is proved that network ASE under distance-based user association serves as the upper bound of that under content-based rule in dense BS regime. To enable more spectrum-efficient user association in dense CSCN, we further optimize network ASE by designing a probabilistic content retrieving strategy based on distance-based rule. With the optimized retrieving probability, network ASE could be substantially enhanced and even increase with the growing BS density in dense BS regime.

Efficient color code decoders in d≥2 dimensions from toric code decoders

We introduce an efficient decoder of the color code in d≥2 dimensions, the Restriction Decoder, which uses any d-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold p2D≈10.2% on the square-octagon lattice is on a par with the toric code threshold on the square lattice.

Sub-Block Rearranged Staircase Codes

We propose a new family of spatially coupled product codes, called sub-block rearranged staircase (SR-staircase) codes. Each code block of SR-staircase codes is obtained by encoding rearranged preceding code blocks and new information blocks, where the rearrangement involves sub-blocks decomposition and transposition. The proposed codes can be constructed to have each code block size of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/q$ </tex-math></inline-formula> to that of the conventional staircase codes while having the same rate and component codes, for any positive integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> . In this regard, we can use strong algebraic component codes to construct SR-staircase codes with a similar or the same code block size and rate as staircase codes with weak component codes. We characterize the decoding threshold of the proposed codes under iterative bounded distance decoding (iBDD) by using density evolution. We also derive the conditions under which they achieve a better decoding threshold than that of staircase codes. Further, we investigate the error floor performance by analyzing the contributing error patterns and their multiplicities. Both theoretical and simulation results show that the designed SR-staircase codes outperform staircase codes in terms of waterfall and error floor while the performance can be further improved by using a large coupling width.

Spatially Coupled PLDPC-Hadamard Convolutional Codes

We propose a new type of ultimate-Shannon-limit-approaching codes called spatially coupled protograph-based low-density parity-check Hadamard convolutional codes (SC-PLDPCH-CCs), which are constructed by spatially coupling PLDPC-Hadamard block codes. We develop an efficient decoding algorithm that combines pipeline decoding and layered scheduling for the decoding of SC-PLDPCH-CCs, and analyze the latency and complexity of the decoder. To estimate the decoding thresholds of SC-PLDPCH-CCs, we first propose a layered protograph extrinsic information transfer (PEXIT) algorithm to evaluate the thresholds of spatially coupled PLDPC-Hadamard terminated codes (SC-PLDPCH-TDCs) with a moderate coupling length. With the use of the proposed layered PEXIT method, we develop a genetic algorithm to find good SC-PLDPCH-TDCs in a systematic way. Then we extend the coupling length of these SC-PLDPCH-TDCs to form good SC-PLDPCH-CCs. Results show that our constructed SC-PLDPCH-CCs can achieve comparable thresholds to the block code counterparts. Simulations illustrate the superiority of the SC-PLDPCH-CCs over the block code counterparts and other state-of-the-art low-rate codes in terms of error performance. For the rate-0.00295 SC-PLDPCH-CC, a bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> is achieved at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E_{b}/N_{0} = -1.465$ </tex-math></inline-formula> dB, which is only 0.125 dB from the ultimate Shannon limit.

Downlink and Uplink Sum Rate Maximization for HAP-LAP Cooperated Networks

Aerial base stations (ABSs) are expected to be important supplementary components for the 5G-and-beyond communication systems to achieve global Internet of Everything. To fully exploit the advantages of the ABSs seamless connection, in this paper, we propose a high altitude platform (HAP) and low altitude platforms (LAPs) cooperated network architecture. Specifically, the HAP can extend the network coverage and LAPs can act as relays to improve the transmission performance for hot spots. In the considered network, we investigate the sum rate maximization problem by optimizing the downlink and the uplink transmissions, respectively. In the downlink, we adopt the orthogonal frequency division multiple access technique and take the basic rate requirement into account. In addition, we optimize the height of LAPs and the spectrum allocation between the HAP and LAPs, and between the LAP and users. Then, we decouple the downlink sum rate maximation problem as three subproblems. An alternating optimization framework is designed to deal with these non-convex optimization problems, where the height and spectrum allocation are tackled in turn. In the uplink, we adopt the non-orthogonal multiple access technique, and consider the decoding threshold of the successive interference cancellation technique. Afterward, we optimize the power allocation of each user, and the spectrum allocation between users and the LAP, and between LAPs and the HAP. Finally, simulation results show the effects of spectrum allocation, height optimization and power allocation on network performance, which verify that the proposed scheme can achieve higher sum rate on both downlink and uplink in comparison with the current works.

Design of Protograph-Based LDPC Codes for Two-User Gaussian Multiple Access Channel

In this letter, we study the performance of protograph-based low-density parity-check (LDPC) codes for two-user Gaussian multiple access channel (GMAC). Considering the characteristic of a joint protograph of two users, we first derive a joint protograph-based extrinsic information transfer (JP-EXIT) chart to analyze the iterative user decoding behavior of two-user GMAC. Guided by the JP-EXIT, we further propose a new design scheme and construct protograph base matrices of different code rates for two users, respectively. Both theoretical analyses and simulation results show that the proposed protograph-based LDPC codes of low decoding threshold outperform the conventional irregular LDPC codes for both equal-power and unequal-power GMAC with different code rates.

Quantum Information Protection Scheme Based on Reinforcement Learning for Periodic Surface Codes

Quantum information transfer is an information processing technology with high speed and high entanglement with the help of quantum mechanics principles. To solve the problem of quantum information getting easily lost during transmission, we choose topological quantum error correction codes as the best candidate codes to improve the fidelity of quantum information. The stability of topological error correction codes brings great convenience to error correction. The quantum error correction codes represented by surface codes have produced very good effects in the error correction mechanism. In order to solve the problem of strong spatial correlation and optimal decoding of surface codes, we introduced a reinforcement learning decoder that can effectively characterize the spatial correlation of error correction codes. At the same time, we use a double-layer convolutional neural network model in the confrontation network to find a better error correction chain, and the generation network can approach the best correction model, to ensure that the discriminant network corrects more nontrivial errors. To improve the efficiency of error correction, we introduced a double-Q algorithm and ResNet network to increase the error correction success rate and training speed of the surface code. Compared with the previous MWPM 0.005 decoder threshold, the success rate has slightly improved, which can reach up to 0.0068 decoder threshold. By using the residual neural network architecture, we saved one-third of the training time and increased the training accuracy to about 96.6%. Using a better training model, we have successfully increased the decoder threshold from 0.0068 to 0.0085, and the depolarized noise model being used does not require a priori basic noise, so that the error correction efficiency of the entire model has slightly improved. Finally, the fidelity of the quantum information has successfully improved from 0.2423 to 0.7423 by using the error correction protection schemes.

Hybrid Coupled Serially Concatenated Codes

In this paper, we present a class of hybrid coupled serially concatenated codes (HC-SCC), in which coupling is achieved by re-encoding partial outputs from both the outer and the inner encoders. Firstly, we derive the exact density evolution (DE) equations over the binary erasure channels (BECs) for the HC-SCCs, which can be used to analyze the impact of the coupling ratios and the coupling memories on the performances of the HC-SCCs. Secondly, we present a genie-aided lower bound for the proposed HC-SCCs to estimate their error-floors. Thirdly, we use the DE equations to design randomly punctured HC-SCCs, including rate-compatible HC-SCCs. The analytical and the numerical results show that: 1) for comparable coupling memories, the iterative decoding thresholds of the proposed HC-SCC ensembles are better than those of the recently introduced spatially coupled serially concatenated code (SC-SCC) ensembles and the partially parity-coupled turbo code (PPC-TC) ensembles; 2) with outer coupling memory five and inner coupling memory five, the iterative decoding thresholds of the rate-compatible HC-SCCs over the BECs are 0.0003 away from the corresponding channel capacities for a wide range of coding rates; 3) simulation results over the additive white Gaussian noise (AWGN) channels and the BECs are provided to confirm the performance advantages of the HC-SCCs. Particularly, the simulation results show that the HC-SCCs admit lower error-floor than the SC-SCCs and the rate-compatible HC-SCCs perform better than the PPC-TCs, which are consistent with the results of DE analysis.

Echo state network based symbol detection in chaotic baseband wireless communication

The Chaotic Baseband Wireless Communication System (CBWCS) is expected to eliminate the Inter-Symbol Interference (ISI) caused by multipath propagation by using the optimal decoding threshold that is the sum of the ISI caused by past decoded bits and the ISI caused by future transmitting bits. However, the current technique is only capable of removing partial effects of the ISI, because only past decoded bits are available for the sub-optimal decoding threshold calculation. The unavailability of the future information needed for the optimal decoding threshold is an obstacle to further improve the Bit Error Rate (BER) performance. In contrast to the previous method using Echo State Network (ESN) to predict one future bit, the proposed method in this paper predicts the optimal decoding threshold directly using ESN. The proposed ESN-based threshold prediction method simplifies the symbol decoding operation by avoiding the iterative prediction of the output waveform points using ESN and accumulated error caused by the iterative operation. With this approach, the calculation complexity is reduced compared to the previous ESN-based approach. The proposed method achieves better BER performance compared to the previous method. The reason for this superior result is twofold. First, the proposed ESN is capable of using more future symbols information conveyed by the ESN input to obtain more accurate threshold rather than the previous method in which only one future symbol was available. Second, the proposed method here does not need to estimate the channel information using Least Squared (LS) method, which avoids the extra error caused by inaccurate channel information estimation. Simulation results and experiment based on a wireless open-access research platform under a practical wireless channel show the effectiveness and superiority of the proposed method.

On the Decoding Performance of Spatially Coupled LDPC Codes With Sub-Block Access

We study spatially coupled LDPC codes that allow access to sub-blocks much smaller than the full code block. Sub-block access is realized by a semi-global decoder that decodes a chosen target sub-block by only accessing the target, plus a prescribed number of helper sub-blocks adjacent in the code chain. This paper develops a theoretical methodology for analyzing the semi-global decoding performance of spatially coupled LDPC codes constructed from protographs. The main result shows that semi-global decoding thresholds can be derived from certain thresholds we define for the single-sub-block graph. These characterizing thresholds are also used for deriving lower bounds on the decoder’s performance over channels with variability across sub-blocks, which are motivated by applications in data storage.

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.

Construction of Multi-Rate Quasi-Cyclic LDPC Codes for Satellite Communications

To provide reliable transmissions with flexible rates for satellite communications, this paper presents a novel method of constructing multi-rate quasi-cyclic low-density parity-check (LDPC) codes. The basic idea is to generate low-rate codes from a high-rate mother code by combining shortening and extending, which ensures that the generated code family owns the same code length, in order to maintain the same frame structure. The code construction involves the design of base matrices and exponent matrices for the designed codes. A progressive row elimination and addition algorithm is proposed for designing the code base matrices from a high rate to low rates. This algorithm leads to the nested and systematic structure of the parity-check matrices, which are desirable for practical implementations of their encoders and decoders, while ensuring the optimal decoding thresholds. In addition, we construct a circulation coefficient matrix based on finite fields and select the optimal rows in this matrix to construct exponent matrices while considering of cycle structures. We demonstrate that the designed codes achieve better performance for all the code rates than the LDPC codes in DVB-S2X standards. In addition, the proposed codes do not exhibit error floors for their block error rates down to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> .