Decoding Complexity Research Articles

Improving Decodability of Polar Codes by Adding Noise

This paper presents an online perturbed and directed neural-evolutionary (Online-PDNE) decoding algorithm for polar codes, in which the perturbation noise and online directed neuro-evolutionary noise sequences are sequentially added to the received sequence for re-decoding if the standard polar decoding fails. The new decoding algorithm converts uncorrectable received sequences into error-correcting regions of their decoding space for correct decoding by adding specific noises. To reduce the decoding complexity and delay, the PDNE decoding algorithm and sole neural-evolutionary (SNE) decoding algorithm for polar codes are further proposed, which provide a considerable tradeoff between the decoding performance and complexity by acquiring the neural-evolutionary noise in an offline manner. Numerical results suggest that our proposed decoding algorithms outperform the other conventional decoding algorithms. At high signal-to-noise ratio (SNR) region, the Online-PDNE decoding algorithm improves bit error rate (BER) performance by more than four orders of magnitude compared with the conventional simplified successive cancellation (SSC) decoding algorithm. Furthermore, in the mid-high SNR region, the average normalized complexity of the proposed algorithm is almost the same as that of the SSC decoding algorithm, while preserving the decoding performance gain.

Sliding Window Decoding for QC-SC-LDPC Codes Under the Constraint of Implementation Complexity

Sliding window decoding has been employed as the decoding scheme of quasi-cyclic spatially-coupled low-density parity-check (QC-SC-LDPC) codes, with the decoding latency and complexity independent of the coupling length but proportional to the window size. The window size was empirically chosen at least three times the constraint length in previous works, which results in the preference of QC-SC-LDPC codes with small constraint length, to reduce the decoding latency and complexity in practical systems. However, the code design freedom and the decoder throughput are limited for these QC-SC-LDPC codes. In this paper, the optimal window size under the constraint of practical decoding complexity is analyzed employing multi-edge-type density evolution (MET-DE). It can be verified from the MET-DE analytical result that the optimal window size could achieve less than twice the constraint length for several edge spreading patterns, which largely loose this restriction in practical code design. Furthermore, it could be observed that besides the constraint length, the specific structure of the edge spreading pattern could affect the optimal window size as well, and thus the MET-DE analytical method for the optimal window size could effectively guide the design of QC-SC-LDPC codes in practical systems. Then an improved sliding window decoding scheme is proposed, including the optimal choice of window size and a parity-check based early termination scheme. Finally, the performance and robustness of the proposed sliding window decoding scheme for QC-SC-LDPC codes are demonstrated via simulation, with the decoding complexity at most twice as that of the conventional LDPC coded schemes.

Secure Private and Adaptive Matrix Multiplication Beyond the Singleton Bound

We consider the problem of designing secure and private codes for distributed matrix-matrix multiplication. A master server owns two private matrices and hires worker nodes to help compute their product. The matrices should remain information-theoretically private from the workers. Some of the workers are malicious and return corrupted results to the master. We design a framework for security against malicious workers in private matrix-matrix multiplication. The main idea is a careful use of Freivalds’ algorithm to detect erroneous matrix multiplications. Our main goal is to apply this security framework to schemes with adaptive rates. Adaptive schemes divide the workers into clusters and thus provide flexibility in trading decoding complexity for efficiency. Our new scheme, SRPM3, provides a computationally efficient security check per cluster that detects the presence of one or more malicious workers with high probability. An additional per worker check is used to identify the malicious nodes. SRPM3 can tolerate the presence of an arbitrary number of malicious workers. We provide theoretical guarantees on the complexity of the security checks and simulation results on both, the missed detection rate as well as on the time needed for the integrity check.

Adaptive Loop Filter Hardware Design for 4K ASIC VVC Decoders

Versatile video coding (VVC) is the next generation video coding standard released in July 2020. VVC introduces new coding tools enhancing the coding efficiency compared to its predecessor high efficiency video coding (HEVC). These new tools have a significant impact on the VVC software decoder with a complexity estimated to two times HEVC decoder complexity. In particular, the adaptive loop filter (ALF) introduced in VVC as an in-loop filter increases both the decoding complexity and memory usage. These concerns need to be carefully addressed regarding the design of an efficient hardware implementation of a VVC decoder. In this paper, we present an efficient hardware implementation of the ALF tool for VVC decoder. The proposed solution establishes a novel scanning order between Luma and Chroma components that reduces significantly the ALF memory. The design takes advantage of all ALF features and establishes an unified hardware module for all ALF filters. The design uses 26 regular multipliers in a pipelined architecture with a fixed throughput of 2 pixels/cycle and fixed system latency regardless of the selected filter. This design operates at 600 MHz frequency enabling to decode on ASIC platform a 4K video at 30 frames per second in 4:2:2 chroma sub-sampling format.

Editorial Real-Time Implementation of VVC Standard for Consumer Electronic Devices

We are pleased to present this Special Issue on this timely and important topic. The next generation ISO/IEC MPEG and ITU-T VCEG video coding standard called versatile video coding (VVC) was released in July 2020 <xref ref-type="bibr" rid="ref1">[1]</xref>, <xref ref-type="bibr" rid="ref2">[2]</xref>. Gls vvc includes some coding tools enabling an important coding gains estimated to 35&#x0025; of bit-rate reductions for the same quality in terms of peak signal to noise ration (PSNR) compared with high efficiency video coding (HEVC) in random access (RA) coding configurations. For all intra (AI) configuration the improvement is a bit lower and estimated in a 25&#x0025;. Moreover, subjective comparison recently conducted between HEVC and the VVC reference software, has shown that this gain is even higher reaching up to 50&#x0025; of bit-rate reduction for the same perceived quality <xref ref-type="bibr" rid="ref3">[3]</xref> - <xref ref-type="bibr" rid="ref4">[4]</xref>. Unfortunately the coding gain is associated to an increase in the algorithm complexity at both encoder and decoder sides. The VVC encoder is estimated to be <inline-formula> <tex-math notation="LaTeX">$8\times $ </tex-math></inline-formula> more complex than HEVC in RA configuration and <inline-formula> <tex-math notation="LaTeX">$25\times $ </tex-math></inline-formula>. The complexity of the VVC decoder is doubled compared to HEVC. Therefore, complexity reduction and optimization of VVC is critical to design of software and hardware real time VVC codecs for successful deployment on video consumer devices including among others TVs, mobile phones and tablets.

Achievable Regions and Precoder Designs for the Multiple Access Wiretap Channels With Confidential and Open Messages

This paper investigates the secrecy achievable region of multiple access wiretap (MAC-WT) channels where, besides confidential messages, the users have also open messages to transmit. All these messages are intended for the legitimate receiver (or Bob for brevity) but only the confidential messages need to be protected from the eavesdropper (Eve). We first consider a discrete memoryless (DM) MAC-WT channel where both Bob and Eve jointly decode their interested messages. By using random coding, we find an achievable rate region, within which perfect secrecy can be realized, i.e., all users can communicate with Bob with arbitrarily small probability of error, while the confidential information leaked to Eve tends to zero. Due to the high implementation complexity of joint decoding, we also consider the DM MAC-WT channel where Bob simply decodes messages independently while Eve still applies joint decoding. We then extend the results in the DM case to a Gaussian vector (GV) MAC-WT channel. Based on the information theoretic results, we further maximize the sum secrecy rate of the GV MAC-WT system by designing precoders for all users. Since the problems are non-convex, we provide iterative algorithms to obtain suboptimal solutions. Simulation results show that compared with existing schemes, secure communication can be greatly enhanced by the proposed algorithms, and in contrast to the works which only focus on the network secrecy performance, the system spectrum efficiency can be effectively improved since open messages can be simultaneously transmitted.

On the Decoding of Lattices Constructed via a Single Parity Check

This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice, named <inline-formula> <tex-math notation="LaTeX">$L_{3\cdot 24}$ </tex-math></inline-formula>, is constructed as a simple application of the single parity check on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check or via the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided.

RS-LDPC stochastic joint decoding algorithm

Due to complexity issues, low-density parity-check (LDPC) codes often use a partial parallel decoding architecture, which cannot easily meet the requirements of high-speed and ultra-high-speed applications. Conversely, the Reed-Solomon (RS) codes often use hard decision decoding algorithms in existing decoding schemes. Therefore, there is room for further optimization in the complexity and performance of concatenated RS -LDPC decoding. In this paper, we propose an RS-LDPC stochastic joint iterative decoding algorithm based on the principle of stochastic computing. This algorithm inherits the advantages of the low complexity of stochastic LDPC decoders and the feature of stochastic RS decoding algorithms, which can use hard decoders. Meanwhile, the decoding performance is improved through the joint iterative decoding architecture. In this respect, we design a test-vector generation method based on LDPC probability values and adopt a novel additional extrinsic information generation mechanism. Such a mechanism is applied to the structure of the variable nodes of the stochastic LDPC decoder. The simulation results show that at a bit error rate of $10^{-6}$, the proposed algorithm can obtain a gain of 0.3 dB as compared with the RS-LDPC cascaded decoding scheme using floating-point belief propagation and Berlekamp -Massey hard decision decoding algorithm. Its implementation complexity is the complexity of a stochastic LDPC decoder and several RS hard decoders. The decoding latency is determined by the latency of the stochastic LDPC decoder. Therefore, the joint iterative decoding algorithm proposed in this paper has a very good application value.

Spatially Coupled Serially Concatenated Codes: Performance Evaluation and VLSI Design Tradeoffs

Spatially coupled serially concatenated codes (SC-SCCs) are constructed by coupling several classical turbo-like component codes. The resulting spatially coupled codes provide a close-to-capacity performance and low error floor, which have attracted a lot of interest in the past few years. The aim of this paper is to perform a comprehensive design space exploration to reveal different aspects of SC-SCCs, which is missing in the literature. More specifically, we investigate the effect of block length, coupling memory, decoding window size, and number of iterations on the decoding performance, complexity, latency, and throughput of SC-SCCs. To this end, we propose two decoding algorithms for the SC-SCCs: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">block-wise</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">window-wise</i> decoders. For these, we present VLSI architectural templates and explore them based on building blocks implemented in 12nm FinFET technology. Linking architectural templates with the new algorithms, we demonstrate various tradeoffs between throughput, silicon area, latency, and decoding performance.

VVC Search Space Analysis Including an Open, Optimized Implementation

Versatile Video Coding (VVC) is a new video coding standard finalized in July 2020. During the standard development much attention was paid to keeping the decoding complexity increase as small as possible, with more permissive approach being taken with regard to the encoding. The VVC reference software VTM in random access configuration requires around double the time to decode and <inline-formula> <tex-math notation="LaTeX">$8\times $ </tex-math></inline-formula> the time to encode a video, compared to High Efficiency Video Coding (HEVC) reference software HM. With this runtime increase, an objective bitrate reduction of around 40&#x0025; is achieved. In this paper we analyze the encoding complexity increase of VVC over HEVC. We abstract the implementation-based aspects, including low-level software optimizations and introduce an empirical measure to quantify the extent of encoder search space given a specific search algorithm. Based on the measure, we compare the search space of HM and VTM, but also of the open and optimized VVC encoder implementation VVenC, showing the potential for search space reduction and its impact on compression performance. Overall, it can be seen that while VVC&#x2019;s search space is quite large in VTM, it can be efficiently limited either by including early termination strategies or by disabling VVC coding tools.

Decoding of Block Codes by using Genetic Algorithms and Permutations Set

Recently Genetic algorithms are successfully used for decoding some classes of error correcting codes. For decoding a linear block code C, these genetic algorithms computes a permutation p of the code generator matrix depending of the received word. Our main contribution in this paper is to choose the permutation p from the automorphism group of C. This choice allows reducing the complexity of re-encoding in the decoding steps when C is cyclic and also to generalize the proposed genetic decoding algorithm for binary nonlinear block codes like the Kerdock codes. In this paper, an efficient stop criterion is proposed and it reduces considerably the decoding complexity of our algorithm. The simulation results of the proposed decoder, over the AWGN channel, show that it reaches the error correcting performances of its competitors. The study of the complexity shows that the proposed decoder is less complex than its competitors that are based also on genetic algorithms.

Decoding of Decode and Forward (DF) Relay Protocol using Min-Sum Based Low Density Parity Check (LDPC) System

Decoding high complexity is a major issue to design a decode and forward (DF) relay protocol. Thus, the establishment of low complexity decoding system would beneficial to assist decode and forward relay protocol. This paper reviews existing methods for the min-sum based LDPC decoding system as the low complexity decoding system. Reference lists of chosen articles were further reviewed for associated publications. This paper introduces comprehensive system model representing and describing the methods developed for LDPC based for DF relay protocol. It is consists of a number of components: (1) encoder and modulation at the source node, (2) demodulation, decoding, encoding and modulation at relay node, and (3) demodulation and decoding at the destination node. This paper also proposes a new taxonomy for min-sum based LDPC decoding techniques, highlights some of the most important components such as data used, result performances and profiles the Variable and Check Node (VCN) operation methods that have the potential to be used in DF relay protocol. Min-sum based LDPC decoding methods have the potential to provide an objective measure the best tradeoff between low complexities decoding process and the decoding error performance, and emerge as a cost-effective solution for practical application.

NARRATIVE ESSENCE OF LANGUAGE CONTENT IN COMPOSERS’ EVALUATION OF THEIR CREATIVITY

The paper investigates the narratives of contemporary Ukrainian composers, in the context of their assessment of their own work and the actual musicological components of each work: its focus, conglomeration of musical expressive means, and their multidimensional and multifunctional context. The paper focuses on the problem of inconsistency of the author’s linguistic content in expressing his own ideas about his work with the actual work in terms of the logic of musicology, psychology, etc. A new narrative approach to the interpretation of musical expressive means has formed in modern compositional work. They are viewed as the combinations of different levels of perception: visual, sensory, linguistic, associative-ambiguous, abstract, etc. Thus, wherever there is an information and communicative environment, there function the complex cognitive formations allowing to reflect and evaluate the imagery of the musical work. Hence, the “conceptual sphere” of each work emerges that consists of the relevant ideas, concepts, images, and statements. The research methodology is based on the analysis of the sphere of expression of human thinking and communicative processes by R. Barthes, C. Bremond, F. Jameson, T. Titarenko, T. Todorov, E. Tshebinsky, and others. Also, the studies by H.-G. Gadamer, M. Heidegger, P. Ricoeur, R. Harré consider the narrative as a discursive structure formed on the basis of personal experience. In the field of musical art, the problem of narratology was addressed in the works of N. Gerasimova-Persidska, O. Zinkevych, Y. Chekan, and some other musicologists who expressed the essence of modern compositional thinking, in line with a new sense of time and space, discreteness and dialogic historical context of culture. To establish the artistic conceptuality of certain musical pieces, the comparative analysis of subjective and authorial understanding of composers of their own works in the musicological, cultural, and socio-political context was made. As the works have a significant number of allusions to the stylistics of different eras, the conflicting emotional reaction of the composers gives grounds to assert the idea of expressing a narrative of existential torment of a torn postmodern mind in the inter-genre web in their works. The paper concludes that assimilation and awareness of meaning are possible only through certain messages that can have many meanings, and complex decoding of them may take any form. Perception of oneself in the social environment occurs through the formation of narratives that express the author’s position. However, this stance voiced in interviews and other language content is not always consistent with the actual music series. Thus, an interpretive understanding of musical creativity emerges at the junction of the narrator’s and the listener’s meanings.

Low-Complexity Chase Decoding of Reed-Solomon Codes Using Channel Evaluation.

A novel time-varying channel adaptive low-complexity chase (LCC) algorithm with low redundancy is proposed, where only the necessary number of test vectors (TVs) are generated and key equations are calculated according to the channel evaluation to reduce the decoding complexity. The algorithm evaluates the error symbol numbers by counting the number of unreliable bits of the received code sequence and dynamically adjusts the decoding parameters, which can reduce a large number of redundant calculations in the decoding process. We provide a simplified multiplicity assignment (MA) scheme and its architecture. Moreover, a multi-functional block that can implement polynomial selection, Chien search and the Forney algorithm (PCF) is provided. On this basis, a high-efficiency LCC decoder with adaptive error-correcting capability is proposed. Compared with the state-of-the-art LCC (TV = 16) decoding, the number of TVs of our decoder was reduced by 50.4% without loss of the frame error rate (FER) performance. The hardware implementation results show that the proposed decoder achieved 81.6% reduced average latency and 150% increased throughput compared to the state-of-the-art LCC decoder.

LDPC decoding with locally informed dynamic scheduling based on the law of large numbers

The informed dynamic scheduling (IDS) strategies, in which the edge message with the maximum message-residual is updated preferentially, achieve remarkable error-correction performance when applied to low-density parity-check (LDPC) codes. However, the IDS strategies incur inferior convergence in iterative decoding owing to the greedy problem, which is called the update-relayed trend here. In order to solve the greediness, two locally informed dynamic scheduling algorithms based on the law of large numbers are proposed. The proposed decoding algorithms use random select of check nodes over a predefined update range (RSPUR) which effectively suppresses the propagation of the update-relayed trend and accordingly restrains the forming of multi-update cycles. Moreover, the decoding algorithm is further improved based on random select of check nodes over an adjustable update range (RSCAR). The update ranges are selected based on the law of large numbers. Therefore, the computational resources can be allocated more equitably by increasing iterations. Simulation results show that both the proposed algorithms achieve excellent performance in terms of throughput and convergence with low decoding complexity over the Additive White Gaussian Noise (AWGN) and the fading channels compared to the previous IDS strategies. Hence, the proposed algorithms behave excellently over the wireless channels.

Coded Distributed Computing With Partial Recovery

Coded computation techniques provide robustness against <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">straggling</i> workers in distributed computing. However, most of the existing schemes require exact provisioning of the straggling behavior and ignore the computations carried out by straggling workers. Moreover, these schemes are typically designed to recover the desired computation results accurately, while in many machine learning and iterative optimization algorithms, faster approximate solutions are known to result in an improvement in the overall convergence time. In this paper, we first introduce a novel coded matrix-vector multiplication scheme, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">coded computation with partial recovery (CCPR)</i> , which benefits from the advantages of both coded and uncoded computation schemes, and reduces both the computation time and the decoding complexity by allowing a trade-off between the accuracy and the speed of computation. We then extend this approach to distributed implementation of more general computation tasks by proposing a coded communication scheme with partial recovery, where the results of subtasks computed by the workers are coded before being communicated. Numerical simulations on a large linear regression task confirm the benefits of the proposed scheme in terms of the trade-off between the computation accuracy and latency.

Self-Adaptive Ordered Statistics Decoder for Finite Block Length Raptor Codes Toward URLLC

Rateless codes can adapt to the wireless channel conditions without accurate channel state information (CSI) at the transmitter side, avoiding CSI feedback and retransmission, and thus are a promising channel coding approach to meet the stringent requirements of ultrareliable low-latency communications (uRLLCs). This article investigates a self-adaptive ordered statistics decoder (S-OSD) scheme for finite block length nonbinary Raptor code (NBRC). Aiming at minimizing the decoding complexity, some strategies for our S-OSD scheme are designed, including the segmentation rules of most reliable basis, the generating and discarding rules of test error patterns, and the stop criteria, respectively. In addition, an upper bound of block error rates (BLERs) for the NBRC under OSD is derived, which can be used to estimate the number of NBRC symbols required to successfully decode the input information via the S-OSD. Simulation results show that the complexity of our S-OSD scheme is greatly reduced comparing to the existing OSD schemes, while achieving very low BLER in the short block length regime.

Deep Neural Network Based Reconciliation for CV-QKD

High-speed reconciliation is indispensable for supporting the continuous-variable quantum key distribution (CV-QKD) system to generate the secure key in real-time. However, the error correction process’s high complexity and low processing speed limit the reconciliation speed. Therefore, reconciliation has also become the bottleneck of system performance. In this paper, we proposed a high-speed reconciliation scheme that uses the deep neural network to optimize the decoding process of the low-density parity-check (LDPC) code. We first introduced a network structure of decoding implementation based on the deep neural network, which can be applied to decoding algorithms of parallel strategy and significantly reduce the decoding complexity. Subsequently, we proposed two improved decoding algorithms based on this structure, named linear fitting algorithm and deep neural network-assisted decoding algorithm. Finally, we introduced a high-speed reconciliation scheme based on the CPU-GPU hybrid platform. Simulation results show that the proposed reconciliation scheme reduces the complexity and enables us to realize the high-speed CV-QKD system. Furthermore, the improved decoding algorithm can also reduce the FER, thereby increasing the secret key rate.

Partial Order-Based Decoding of Rate-1 Nodes in Fast Simplified Successive-Cancellation List Decoders for Polar Codes

Polar codes are the first family of error-correcting codes that can achieve channel capacity. Among the known decoding algorithms, Successive-Cancellation List (SCL) decoding supported by a Cyclic Redundancy Check (CRC) shows the best error-correction performance at the cost of a high decoding complexity. The decoding of Rate-1 nodes belongs to the most complex tasks in SCL decoding. In this paper, we present a new algorithm that largely reduces the number of considered candidates in a Rate-1 node and generate all required candidates in parallel. For this purpose, we use a partial order of the candidate paths to prove that only a specified number of candidates needs to be considered. Further complexity reductions are achieved by an extended threshold-based path exclusion scheme at the cost of negligible error-correction performance loss. We present detailed Application-Specific Integrated Circuit (ASIC) implementation data on a 28 nm Fully Depleted Silicon on Insulator (FD-SOI) Complementary Metal-Oxide-Semiconductor (CMOS) technology for decoders with code length 128. We show that the new decoders outperform state-of-the-art reference decoders. For list size 8, improvements of up to 158.8% and 62.5% in area and energy efficiency are observed, respectively.

THINK: A novel conversation model for generating grammatically correct and coherent responses

Many existing conversation models that are based on the encoder–decoder framework incorporate complex encoders. These powerful encoders serve to enrich the context vectors, so that the generated responses are more diverse and informative. However, these approaches face two potential challenges. First, the high complexity of the encoder means relative simplicity of the decoder. There is a danger that the decoder becomes too simple to effectively capture previously generated information. As a result, the decoder may produce duplicated and self-contradicting responses. Second, by having a complex encoder, the model may generate incoherent responses because the complex context vectors may deviate from the true semantics of context. In this work, we propose a conversation model named “THINK” (Teamwork generation Hover around Impressive Noticeable Keywords) that is equipped with a complex decoder to avoid generating duplicated and self-contradicting responses. The model also simplifies the context vectors and increases the coherence of generated responses in a reasonable way. For this model, we propose Teamwork generation framework and Semantics extractor. Compared with other baselines, both automatic and human evaluation showed the advantages of our model.