Decoding Graph Research Articles

Techniques for combining fast local decoders with global decoders under circuit-level noise

Implementing algorithms on a fault-tolerant quantum computer will require fast decoding throughput and latency times to prevent an exponential increase in buffer times between the applications of gates. In this work we begin by quantifying these requirements. We then introduce the construction of local neural network (NN) decoders using three-dimensional convolutions. These local decoders are adapted to circuit-level noise and can be applied to surface code volumes of arbitrary size. Their application removes errors arising from a certain number of faults, which serves to substantially reduce the syndrome density. Remaining errors can then be corrected by a global decoder, such as Blossom or union find, with their implementation significantly accelerated due to the reduced syndrome density. However, in the circuit-level setting, the corrections applied by the local decoder introduce many vertical pairs of highlighted vertices. To obtain a low syndrome density in the presence of vertical pairs, we consider a strategy of performing a syndrome collapse which removes many vertical pairs and reduces the size of the decoding graph used by the global decoder. We also consider a strategy of performing a vertical cleanup, which consists of removing all local vertical pairs prior to implementing the global decoder. By applying our local NN decoder and the vertical cleanup strategy to a d = 17 surface code volume, we show a speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.

Average Packet Decoding Delay Minimization for Rate-Aware Buffered Instantly Decodable Network Codes

Buffered instantly decodable network coding (B-IDNC) can cache the non-instantly decodable network packets (NIDPs) containing two wanted packets for subsequent network decoding. This letter proposes a cross-layer scheme named as rate-aware B-IDNC (RA-B-IDNC) by jointly considering the B-IDNC decision at the network layer and transmission rate adaption at the physical layer to minimize the average packet decoding delay. Due to the intractability of the considered problem, we approximate it as maximizing the recovery rate which is defined as the ratio of the total number of instantly decoded packets to the transmission time. We design two graphs named as RA-B-IDNC encoding graph which reflects the encoding relations among the vertices of the graph, and RA-B-IDNC decoding graph which indicates the decoding relations among the wanted packets of each receiving node by exploiting its buffered NIDPs. Using these graphs, the approximated problem can be formulated as a maximum weight encoding clique problem. Simulation results demonstrate that RA-B-IDNC can reduce the average packet decoding delay compared with the existing algorithms in the literature.

Higher-Order Belief Propagation Correction Decoder for Polar Codes.

Belief propagation (BP) decoding for polar codes has been extensively studied because of its inherent parallelism. However, its performance remains inferior to that of successive cancellation list decoding (SCL) due to the structure of the decoding graph. To improve the block error rate (BLER) performance, the BP correction (BPC) decoding, a post-processing scheme that corrects prior knowledge of the identified code bit, improves convergence by executing additional iterations on the failed BP decoder. Moreover, the BPC decoder demonstrates a better decoding performance than the BP-based bit-flipping decoder. Nevertheless, the additional decoding attempts lead to increased latency. In this article, a modified BPC decoder is proposed to reduce the number of decoding attempts by redefining the correction rules. A new metric is designed to effectively identify the corrected location. Numerical results show that the proposed modified BPC decoder achieves a slight improvement in BLER compared with the original BPC, with a dramatic reduction in average complexity. Furthermore, a higher-order version, named MBPC-, is extended to further improve the performance, where the is the maximum correction order. Numerical results show that the higher-order modified BPC achieves a similar BLER performance to existing multiple bit-flipping BP decoders but has around half the latency overhead. In addition, the proposed MBPC-2 decoder performs better than the cyclic redundancy check-aided SCL (CA-SCL) decoder with list size 4 and is slightly worse than the CA-SCL with list size 8 in high signal-to-noise ratio (SNR) regions but with significant decoding latency reduction.

Stopping-Set Assisted Reinforced Belief Propagation for Decoding Short LDPC Codes

The low-density parity-check (LDPC) code is chosen to be a channel coding scheme for ultra-reliable and low-latency communications (URLLC) in 5G. However, short LDPC compromises the belief propagation (BP) decoder because of the stopping set (SS) in the decoding graph. We propose to perform different node operations according to the SS. A dynamic SS selection according to the received sequence is further applied. In the length 308 5G LDPC, our decoders outperform prior work, improving BP and collaborative BP at block error probability (BLER) of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3 \times 10^{-5}$ </tex-math></inline-formula> by around 0.176 and 0.092 dB respectively, with almost the same number of iterations.

A Novel Human-Vehicle Interaction Assistive Device for Arab Drivers Using Speech Recognition

About one-quarter of all car collisions in the United States are caused by distracted driving, and this ratio is expected to rise. As vehicles are equipped with more elaborate and complex technology, human-vehicle interaction via dashboard displays and controls will become more complex and distracting. Human-vehicle interaction via voice-based technology offers a less distracting alternative. In this study we aim to develop a voice-based car assistant, with a focus on Arabic language speech recognition. We prepare a new 4000-word domain-specific lexicon to comprehensively support driver-vehicle interactions, and we create corresponding text and speech corpora. Then we extract acoustic feature vectors and use various acoustic models to support speech recognition. The language model is created using an n-gram model. Then acoustic and language models, and the lexicon are combined to generate a decoding graph. The text corpus consists of 6110 elements, including words, phrases, and sentences. The speech corpus has more than 60000 recordings (almost 50 hours). For the decoding of noise-free audio, a Deep Neural Network + Hidden Markov Model provided 94.832% accuracy, a Subspace Gaussian Mixture Model + Hidden Markov Model provided 94.2% accuracy, and the best Gaussian Mixture Model + Hidden Markov Model provided 94.13% accuracy. For the decoding of noisy audio, a Deep Neural Network + Hidden Markov Model provided 93.316% accuracy, a Subspace Gaussian Mixture Model + Hidden Markov Model provided 92.62% accuracy, and the best Gaussian Mixture Model + Hidden Markov Model provided 91.82% accuracy. A usability study was conducted on the system with 10 participants. Almost all of the results of that study showed usability ratings of greater than 4.0 out of 5.0. These usability ratings indicate that the proposed system was seen by the participants as important, and useful for reducing driver distraction.

Robust voice user interface for internet-of-things

Internet of things (IoT) plays significant role in the fourth industrial revolution and attracts an increasing interest due to the rapid development of smart devices. IoT comprises factors of twofold. Firstly, a set of things (i.e., appliances, devices, vehicles, etc.) connected together via network. Secondly, human-device interaction to communicate with these things. Speech is the most natural methodology of interaction that can enrich user experience. In this paper, we propose a novel and effective approach for building customized voice interaction for controlling smart devices in IoT environments (i.e., Smart home). The proposed approach is based on extracting customized tiny decoding graph from a large graph constructed using weighted finite sates transducers. Experimental results showed that tiny decoding graphs are very efficient in terms of computational resources and recognition accuracy in clean and noisy conditions. To emphasize the effectiveness of the proposed approach, the standard Resources Management (RM1) dataset was employed and promising results were achieved when compared with four competitive approaches.

A Graph-Theoretical Method for Decoding Some Group MLD-Codes

We construct the class of majority-logical decodable group codes using a method for combining the codes that are based on the tensor product and the sum of codes. The construction of this class rests on the Kasami–Lin technique, which allows us to consider not only individual codes but also families of codes and utilizes the $$M$$ -orthogonality construction presented by Massey that is important for the majority-logical decodable codes. The codes under study are ideals in group algebras over, generally speaking, noncommutative finite groups. Some algorithmic model of the majority decoding for the codes under consideration is developed that is based on the graph-theoretic approach. An important part of this model is the construction of a special decoding graph for decoding one coordinate of a noisy codeword corresponding to this graph. The group properties of the codes enable us to quickly find decoding graphs for the remaining coordinates. We develop some decoding algorithm that corrects the errors in all coordinates of the noisy codeword using this decoding graphs. As an example of families of group codes, we give the Reed–Muller binary codes important in cryptography. The code cryptosystems are considered as an alternative to the number-theoretic cryptosystems widely used at present since they are resistant to attacks by quantum computers. The relevance of the problem under consideration lies in the fact that the use of group codes and their various combinations is currently one of the promising ways to increase the stability of code cryptosystems because enables us to construct new codes with a complex algebraic structure, which positively affects the stability of the code cryptosystems.

On Design of Protograph LDPC Codes for Large-Scale MIMO Systems

Protograph low-density parity-check (LDPC) codes and large-scale multi-input multi-output (LS-MIMO) systems have achieved great interest with various practical applications. However, how to effectively evaluate and design protograph LDPC codes for LS-MIMO systems remains a challenging yet critical problem, especially for low-latency applications. To solve that design challenge, the protograph extrinsic information transfer chart (PEXIT) algorithm for LS-MIMO systems, so-called LS-MIMO-PEXIT algorithm, is first derived based on the mutual information functions of messages that are passed on the joint MIMO detection and LDPC decoding graph. The proposed LS-MIMO-PEXIT algorithm plays a vital role in the optimization process of designing new protograph LDPC codes, tailored for LS-MIMO communications systems. Experiment results demonstrate that the analytical results based on the LS-MIMO-PEXIT algorithm are in good agreement with the simulation results under various input constraints, including the coding rate, the number of decoding iterations, and the LS-MIMO configuration. On top of that, the new protograph LDPC codes designed using our LS-MIMO-PEXIT algorithm achieve a coding gain from 0.2 dB at a low coding rate to 0.4 dB at a high coding rate in comparison with the state-of-the-art protograph codes in the literature. Additionally, we incorporate the practical design experience and the theoretical analysis of mutual functions into a two-step procedure to search for protograph LDPC codes that do not have error-floor behavior at frame error rate (FER) or bit error rate (BER) as low as 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> or 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-7</sup> , respectively. Together with the coding gain, the error-floor-free feature of the proposed protograph LDPC codes is vitally important for future wireless networks where the ultra-reliability is one of the critical requirements.

Improved online fountain codes

Online fountain codes have been proven to require lower overhead and fewer feedbacks than growth codes for successful decoding. In an attempt to improve the intermediate symbol recovery rate, the authors propose sending a number of degree-1 input symbols prior to the build-up phase based on a simple application of probability theory. In addition, during the completion phase, received encoded symbols with three neighbouring white (un-decoded) symbols are retained for decoding and updating the decoding graph later. The performance and characteristics of the proposed improved online fountain codes are compared to those of the original online fountain codes over an erasure channel. Simulation results reveal that the improved online fountain codes outperform the original fountain codes in terms of intermediate symbol recovery rate, average encoded symbols required to be generated by the sender, average feedback transmissions, and encoding/decoding efficiency.

Improved online fountain codes based on shaping for left degree distribution

In this paper, an improved encoding scheme for online fountain codes is proposed with the joint optimization of variable node degree and check node degree is proposed. The coding scheme can be divided into the build-up phase and the completion phase. In the build-up phase, left degree distribution is exploited to guarantee optimal performance phase by modifying the traditional coding scheme of choosing input symbols uniformly at random. A memory-based selecting of the source symbols is employed to decrease the number of connected components, which can thus produce the dimension increasement of the linear subspace of a decoding graph constructed in the build-up phase. The upper bound on coding overhead is also derived from the analysis of random graph theory. Compared with conventional online fountain codes, it can be seen from the simulation results that the proposed scheme can provide significant performance improvement with respect to both coding overhead and feedback cost. Moreover, the lower encoding/decoding complexities may make the proposed scheme more practical in energy-constrained applications such as distributed storage.

Erasure Floor Analysis of Distributed LT Codes

We investigate the erasure floor performance of distributed Luby transform (DLT) codes for transmission within a multi-source, single-relay, and single-destination erasure-link network. In general, Luby transform (LT) codes exhibit a high erasure floor due to poor minimum-distance properties, which can be improved by maximizing the minimum variable-node degree. The same behavior is observed for DLT codes, and therefore a new combining scheme at the relay is proposed to maximize the minimum variable-node degree in the decoding graph. Furthermore, the encoding process at the sources and the combining scheme at the relay are coordinated to improve the transmission overhead. To characterize the asymptotic performance of the proposed DLT codes, we derive closed-form density-evolution expressions, considering both lossless and lossy source-relay channels, respectively. To support the asymptotic analysis, we evaluate the performance of the proposed DLT codes by numerical examples and demonstrate that the numerical results correspond closely to the analysis. Significant improvements in both the erasure floor and transmission overhead are obtained for the proposed DLT codes, as compared to conventional DLT codes.

Improved Low-Density Parity Check Accumulate (LDPCA) Codes

We present improved constructions for Low-Density Parity-Check Accumulate (LDPCA) codes, which are rate-adaptive codes commonly used for distributed source coding (DSC) applications. Our proposed constructions mirror the traditional LDPCA approach; higher rate codes are obtained by splitting the check nodes in the decoding graph of lower rate codes, beginning with a lowest rate mother code. In a departure from the uniform splitting strategy adopted by prior LDPCA codes, however, the proposed constructions introduce non-uniform splitting of the check nodes at higher rates. Codes are designed by a global minimization of the average rate gap between the code operating rates and the corresponding theoretical lower bounds evaluated by density-evolution. In the process of formulating the design framework, the paper also contributes a formal definition of LDPCA codes. Performance improvements provided by the proposed non-uniform splitting strategy over the conventional uniform splitting approach used in prior work are substantiated via density evolution based analysis and DSC codec simulations. Optimized designs for our proposed constructions yield codes with a lower average rate gap than conventional designs and alleviate the trade-off between the performance at different rates inherent in conventional designs. A software implementation is provided for the codec developed.

Two Informed Dynamic Scheduling Strategies for Iterative LDPC Decoders

When residual belief-propagation (RBP), which is a kind of informed dynamic scheduling (IDS), is applied to low-density parity-check (LDPC) codes, the convergence speed in error-rate performance can be significantly improved. However, the RBP decoders presented in previous literature suffer from poor convergence error-rate performance due to the two phenomena explored in this paper. The first is the greedy-group phenomenon, which results in a small part of the decoding graph occupying most of the decoding resources. By limiting the number of updates for each edge message in the decoding graph, the proposed Quota-based RBP (Q-RBP) schedule can reduce the probability of greedy groups forming. The other phenomenon is the silent-variable-nodes issue, which is a condition where some variable nodes have no chance of contributing their intrinsic messages to the decoding process. As a result, we propose the Silent-Variable-Node-Free RBP (SVNF-RBP) schedule, which can force all variable nodes to contribute their intrinsic messages to the decoding process equally. Both the Q-RBP and the SVNF-RBP provide appealing convergence speed and convergence error-rate performance compared to previous IDS decoders for both dedicated and punctured LDPC codes.

Pilotless Frame Synchronization via LDPC Code Constraint Feedback

In traditional receiver architectures, frame synchronization is performed using pilot symbols and a correlation rule. In this paper we show that outputs from the constraint node side of a bipartite decoding graph can be used to achieve frame synchronization in a pilotless low-density parity-check (LDPC) coded transmission, thereby avoiding the bandwidth cost inherent in use of pilot symbols. The complexity of the frame synchronizer is kept relatively low due to its XOR-based approach.

An analysis of belief propagation on the turbo decoding graph with Gaussian densities

Motivated by its success in decoding turbo codes, we provide an analysis of the belief propagation algorithm on the turbo decoding graph with Gaussian densities. In this context, we are able to show that, under certain conditions, the algorithm converges and that-somewhat surprisingly-though the density generated by belief propagation may differ significantly from the desired posterior density, the means of these two densities coincide. Since computation of posterior distributions is tractable when densities are Gaussian, use of belief propagation in such a setting may appear unwarranted. Indeed, our primary motivation for studying belief propagation in this context stems from a desire to enhance our understanding of the algorithm's dynamics in a non-Gaussian setting, and to gain insights into its excellent performance in turbo codes. Nevertheless, even when the densities are Gaussian, belief propagation may sometimes provide a more efficient alternative to traditional inference methods.

The distribution of loop lengths in graphical models for turbo decoding

This paper analyzes the distribution of cycle lengths in turbo decoding and low-density parity check (LDPC) graphs. The properties of such cycles are of significant interest in the context of iterative decoding algorithms which are based on belief propagation or message passing. We estimate the probability that there exist no simple cycles of length less than or equal to k at a randomly chosen node in a turbo decoding graph using a combination of counting arguments and independence assumptions. For large block lengths n, this probability is approximately e^{-{2^{k-1}-4}/n}, k>=4. Simulation results validate the accuracy of the various approximations. For example, for turbo codes with a block length of 64000, a randomly chosen node has a less than 1% chance of being on a cycle of length less than or equal to 10, but has a greater than 99.9% chance of being on a cycle of length less than or equal to 20. The effect of the "S-random" permutation is also analyzed and it is shown that while it eliminates short cycles of length k<8, it does not significantly affect the overall distribution of cycle lengths. Similar analyses and simulations are also presented for graphs for LDPC codes. The paper concludes by commenting briefly on how these results may provide insight into the practical success of iterative decoding methods.