In this paper,an efficient unequal error protection(UEP)scheme for online fountain codes is proposed.In the buildup phase,the traversing-selection strategy is proposed to select the most important symbols(MIS).Then,in...In this paper,an efficient unequal error protection(UEP)scheme for online fountain codes is proposed.In the buildup phase,the traversing-selection strategy is proposed to select the most important symbols(MIS).Then,in the completion phase,the weighted-selection strategy is applied to provide low overhead.The performance of the proposed scheme is analyzed and compared with the existing UEP online fountain scheme.Simulation results show that in terms of MIS and the least important symbols(LIS),when the bit error ratio is 10-4,the proposed scheme can achieve 85%and 31.58%overhead reduction,respectively.展开更多
Quantum error correction, a technique that relies on the principle of redundancy to encode logical information into additional qubits to better protect the system from noise, is necessary to design a viable quantum co...Quantum error correction, a technique that relies on the principle of redundancy to encode logical information into additional qubits to better protect the system from noise, is necessary to design a viable quantum computer. For this new topological stabilizer code-XYZ^(2) code defined on the cellular lattice, it is implemented on a hexagonal lattice of qubits and it encodes the logical qubits with the help of stabilizer measurements of weight six and weight two. However topological stabilizer codes in cellular lattice quantum systems suffer from the detrimental effects of noise due to interaction with the environment. Several decoding approaches have been proposed to address this problem. Here, we propose the use of a state-attention based reinforcement learning decoder to decode XYZ^(2) codes, which enables the decoder to more accurately focus on the information related to the current decoding position, and the error correction accuracy of our reinforcement learning decoder model under the optimisation conditions can reach 83.27% under the depolarizing noise model, and we have measured thresholds of 0.18856 and 0.19043 for XYZ^(2) codes at code spacing of 3–7 and 7–11, respectively. our study provides directions and ideas for applications of decoding schemes combining reinforcement learning attention mechanisms to other topological quantum error-correcting codes.展开更多
Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error corre...Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models.In this paper,we use a distributed decoding strategy,which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases.Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder.The recurrent neural network is trained through the ResNet architecture to improve its decoding accuracy.Then we test the decoding performance of our distributed strategy decoder,recurrent neural network decoder,and the classic minimum weight perfect matching(MWPM)decoder for rotated surface codes with different code distances under the circuit noise model,the thresholds of these three decoders are about 0.0052,0.0051,and 0.0049,respectively.Our results demonstrate that the distributed strategy decoder outperforms the other two decoders,achieving approximately a 5%improvement in decoding efficiency compared to the MWPM decoder and approximately a 2%improvement compared to the recurrent neural network decoder.展开更多
Belief propagation list(BPL) decoding for polar codes has attracted more attention due to its inherent parallel nature. However, a large gap still exists with CRC-aided SCL(CA-SCL) decoding.In this work, an improved s...Belief propagation list(BPL) decoding for polar codes has attracted more attention due to its inherent parallel nature. However, a large gap still exists with CRC-aided SCL(CA-SCL) decoding.In this work, an improved segmented belief propagation list decoding based on bit flipping(SBPL-BF) is proposed. On the one hand, the proposed algorithm makes use of the cooperative characteristic in BPL decoding such that the codeword is decoded in different BP decoders. Based on this characteristic, the unreliable bits for flipping could be split into multiple subblocks and could be flipped in different decoders simultaneously. On the other hand, a more flexible and effective processing strategy for the priori information of the unfrozen bits that do not need to be flipped is designed to improve the decoding convergence. In addition, this is the first proposal in BPL decoding which jointly optimizes the bit flipping of the information bits and the code bits. In particular, for bit flipping of the code bits, a H-matrix aided bit-flipping algorithm is designed to enhance the accuracy in identifying erroneous code bits. The simulation results show that the proposed algorithm significantly improves the errorcorrection performance of BPL decoding for medium and long codes. It is more than 0.25 d B better than the state-of-the-art BPL decoding at a block error rate(BLER) of 10^(-5), and outperforms CA-SCL decoding in the low signal-to-noise(SNR) region for(1024, 0.5)polar codes.展开更多
Power line communication(PLC)has the potential to become the preferred technique for providing broadband communication to homes and offices with advantage of eliminating the need for new wiring infrastructure and redu...Power line communication(PLC)has the potential to become the preferred technique for providing broadband communication to homes and offices with advantage of eliminating the need for new wiring infrastructure and reducing the cost.But it suffers from the impulsive noise because it introduces significant time variance into the power line channel.In this paper,a polar codes based orthogonal frequency division multiplexing(OFDM)PLC system is proposed to deal with the impulsive noise and thereby improve the transmission performance.Firstly,the impulsive noise is modelled with a multi-damped sine function by analyzing the time behavior of impulse events.Then the polar codes are used to combat the impulsive noise of PLC channel,and a low complexity bit-flipping decoding method based on CRC-aided successive cancellation list(CA-SCL)decoding algorithm is proposed.Simulations evaluate the proposed decoding algorithm and the results validate the suggested polar codes based OFDM-PLC scheme which can improve the BER performance of PLC with impulsive interference.展开更多
Topology optimization(TO),a numerical technique to find the optimalmaterial layoutwith a given design domain,has attracted interest from researchers in the field of structural optimization in recent years.For beginner...Topology optimization(TO),a numerical technique to find the optimalmaterial layoutwith a given design domain,has attracted interest from researchers in the field of structural optimization in recent years.For beginners,opensource codes are undoubtedly the best alternative to learning TO,which can elaborate the implementation of a method in detail and easily engage more people to employ and extend the method.In this paper,we present a summary of various open-source codes and related literature on TO methods,including solid isotropic material with penalization(SIMP),evolutionary method,level set method(LSM),moving morphable components/voids(MMC/MMV)methods,multiscale topology optimization method,etc.Simultaneously,we classify the codes into five levels,fromeasy to difficult,depending on their difficulty,so that beginners can get started and understand the form of code implementation more quickly.展开更多
After the pursuit of seventy years,the invention of polar codes indicates that we have found the first capacity-achieving coding with low complexity construction and decoding,which is the great breakthrough of the cod...After the pursuit of seventy years,the invention of polar codes indicates that we have found the first capacity-achieving coding with low complexity construction and decoding,which is the great breakthrough of the coding theory in the past two decades.In this survey,we retrospect the history of polar codes and summarize the advancement in the past ten years.First,the primary principle of channel polarization is investigated such that the basic construction,coding method and the classic successive cancellation(SC)decoding are reviewed.Second,in order to improve the performance of the finite code length,we introduce the guiding principle and conclude five design criteria for the construction,design and implementation of the polar code in the practical communication system based on the exemplar schemes in the literature.Especially,we explain the design principle behind the concatenated coding and rate matching of polar codes in 5G wireless system.Furthermore,the improved SC decoding algorithms,such as SC list(SCL)decoding and SC stack(SCS)decoding etc.,are investigated and compared.Finally,the research prospects of polar codes for the future 6G communication system are explored,including the optimization of short polar codes,coding construction in fading channels,polar coded modulation and HARQ,and the polar coded transmission,namely polar processing.Predictably,as a new coding methodology,polar codes will shine a light on communication theory and unveil a revolution in transmission technology.展开更多
Recently,a generalized successive cancellation list(SCL)decoder implemented with shiftedpruning(SP)scheme,namely the SCL-SP-ωdecoder,is presented for polar codes,which is able to shift the pruning window at mostωtim...Recently,a generalized successive cancellation list(SCL)decoder implemented with shiftedpruning(SP)scheme,namely the SCL-SP-ωdecoder,is presented for polar codes,which is able to shift the pruning window at mostωtimes during each SCL re-decoding attempt to prevent the correct path from being eliminated.The candidate positions for applying the SP scheme are selected by a shifting metric based on the probability that the elimination occurs.However,the number of exponential/logarithm operations involved in the SCL-SP-ωdecoder grows linearly with the number of information bits and list size,which leads to high computational complexity.In this paper,we present a detailed analysis of the SCL-SP-ωdecoder in terms of the decoding performance and complexity,which unveils that the choice of the shifting metric is essential for improving the decoding performance and reducing the re-decoding attempts simultaneously.Then,we introduce a simplified metric derived from the path metric(PM)domain,and a custom-tailored deep learning(DL)network is further designed to enhance the efficiency of the proposed simplified metric.The proposed metrics are both free of transcendental functions and hence,are more hardware-friendly than the existing metrics.Simulation results show that the proposed DL-aided metric provides the best error correction performance as comparison with the state of the art.展开更多
To remove the restriction on code length of polar codes,this paper proposes a construction scheme,called stepwise polar codes,which can gen-erate arbitrary-length polar codes.The stepwise polar codes are generated by ...To remove the restriction on code length of polar codes,this paper proposes a construction scheme,called stepwise polar codes,which can gen-erate arbitrary-length polar codes.The stepwise polar codes are generated by sub-polar codes with different code lengths.To improve coding performance,sub-polar codes are united by polarization effect priority algorithm,which can reduce the number of in-completely polarized channels.Then,the construction method of the generator matrix of the stepwise po-lar code is presented.Furthermore,we prove that the proposed scheme has lower decoding complexity than punctured,multi-kernel polar codes.Simulation results show that the proposed method can achieve similar decoding performance compared with the conventional punctured polar codes,rate-compatible punctured polar code,PC-short and asymmetric polar codes(APC)when code length N=48 and 72,respectively.展开更多
Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, ...Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.展开更多
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 manua...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 protograph 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.展开更多
In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the s...In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the Mac Williams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.展开更多
Belief propagation(BP)decoding outputs soft information and can be naturally used in iterative receivers.BP list(BPL)decoding provides comparable error-correction performance to the successive cancellation list(SCL)de...Belief propagation(BP)decoding outputs soft information and can be naturally used in iterative receivers.BP list(BPL)decoding provides comparable error-correction performance to the successive cancellation list(SCL)decoding.In this paper,we firstly introduce an enhanced code construction scheme for BPL decoding to improve its errorcorrection capability.Then,a GPU-based BPL decoder with adoption of the new code construction is presented.Finally,the proposed BPL decoder is tested on NVIDIA RTX3070 and GTX1060.Experimental results show that the presented BPL decoder with early termination criterion achieves above 1 Gbps throughput on RTX3070 for the code(1024,512)with 32 lists under good channel conditions.展开更多
As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper,...As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.展开更多
The error correction performance of Belief Propagation(BP)decoding for polar codes is satisfactory compared with the Successive Cancellation(SC)decoding.Nevertheless,it has to complete a fixed number of iterations,whi...The error correction performance of Belief Propagation(BP)decoding for polar codes is satisfactory compared with the Successive Cancellation(SC)decoding.Nevertheless,it has to complete a fixed number of iterations,which results in high computational complexity.This necessitates an intelligent identification of successful BP decoding for early termination of the decoding process to avoid unnecessary iterations and minimize the computational complexity of BP decoding.This paper proposes a hybrid technique that combines the“paritycheck”with the“G-matrix”to reduce the computational complexity of BP decoder for polar codes.The proposed hybrid technique takes advantage of the parity-check to intelligently identify the valid codeword at an early stage and terminate the BP decoding process,which minimizes the overhead of the G-matrix and reduces the computational complexity of BP decoding.We explore a detailed mechanism incorporating the parity bits as outer code and prove that the proposed hybrid technique minimizes the computational complexity while preserving the BP error correction performance.Moreover,mathematical formulation for the proposed hybrid technique that minimizes the computation cost of the G-matrix is elaborated.The performance of the proposed hybrid technique is validated by comparing it with the state-of-the-art early stopping criteria for BP decoding.Simulation results show that the proposed hybrid technique reduces the iterations by about 90%of BP decoding in a high Signal-to-Noise Ratio(SNR)(i.e.,3.5~4 dB),and approaches the error correction performance of G-matrix and conventional BP decoder for polar codes.展开更多
Low-density parity-check(LDPC)codes are not only capacity-approaching,but also greatly suitable for high-throughput implementation.Thus,they are the most popular codes for high-speed data transmission in the past two ...Low-density parity-check(LDPC)codes are not only capacity-approaching,but also greatly suitable for high-throughput implementation.Thus,they are the most popular codes for high-speed data transmission in the past two decades.Thanks to the low-density property of their parity-check matrices,the optimal maximum a posteriori probability decoding of LDPC codes can be approximated by message-passing decoding with linear complexity and highly parallel nature.Then,it reveals that the approximation has to carry on Tanner graphs without short cycles and small trapping sets.Last,it demonstrates that well-designed LDPC codes with the aid of computer simulation and asymptotic analysis tools are able to approach the channel capacity.Moreover,quasi-cyclic(QC)structure is introduced to significantly facilitate their high-throughput implementation.In fact,compared to the other capacity-approaching codes,QC-LDPC codes can provide better area-efficiency and energy-efficiency.As a result,they are widely applied in numerous communication systems,e.g.,Landsat satellites,Chang’e Chinese Lunar mission,5G mobile communications and so on.What’s more,its extension to non-binary Galois fields has been adopted as the channel coding scheme for BeiDou navigation satellite system.展开更多
For polar codes,the performance of successive cancellation list(SCL)decoding is capable of approaching that of maximum likelihood decoding.However,the existing hardware architectures for the SCL decoding suffer from h...For polar codes,the performance of successive cancellation list(SCL)decoding is capable of approaching that of maximum likelihood decoding.However,the existing hardware architectures for the SCL decoding suffer from high hardware complexity due to calculating L decoding paths simultaneously,which are unfriendly to the devices with limited logical resources,such as field programmable gate arrays(FPGAs).In this paper,we propose a list-serial pipelined hardware architecture with low complexity for the SCL decoding,where the serial calculation and the pipelined operation are elegantly combined to strike a balance between the complexity and the latency.Moreover,we employ only one successive cancellation(SC)decoder core without L×L crossbars,and reduce the number of inputs of the metric sorter from 2L to L+2.Finally,the FPGA implementations show that the hardware resource consumption is significantly reduced with negligible decoding performance loss.展开更多
Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers.In order to solve the problem of influence of errors on physical qubits,we propose an approximat...Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers.In order to solve the problem of influence of errors on physical qubits,we propose an approximate error correction scheme that performs dimension mapping operations on surface codes.This error correction scheme utilizes the topological properties of error correction codes to map the surface code dimension to three dimensions.Compared to previous error correction schemes,the present three-dimensional surface code exhibits good scalability due to its higher redundancy and more efficient error correction capabilities.By reducing the number of ancilla qubits required for error correction,this approach achieves savings in measurement space and reduces resource consumption costs.In order to improve the decoding efficiency and solve the problem of the correlation between the surface code stabilizer and the 3D space after dimension mapping,we employ a reinforcement learning(RL)decoder based on deep Q-learning,which enables faster identification of the optimal syndrome and achieves better thresholds through conditional optimization.Compared to the minimum weight perfect matching decoding,the threshold of the RL trained model reaches 0.78%,which is 56%higher and enables large-scale fault-tolerant quantum computation.展开更多
In this paper,we propose an arbitrary decode-forward single-relay scheme for finite blocklength polar codes,which can be applied to the general symmetric discrete memoryless relay channel with orthogonal receiver comp...In this paper,we propose an arbitrary decode-forward single-relay scheme for finite blocklength polar codes,which can be applied to the general symmetric discrete memoryless relay channel with orthogonal receiver components.The relay node decodes the received message.The relay node selectively re-encodes the message and transmits it to the destination node.Furthermore,in order to minimize the upper-bound of the block error probability,we propose a selection strategy to decide the proper re-encoded bit set by the relay.Simulation results are presented to illustrate the improvement in decoding performance of the proposed scheme compared to conventional relay schemes in both additive white Gaussian noise(AWGN)channel and Rayleigh fading channel(RFC).展开更多
基金supported by the National Natural Science Foundation of China(61601147)the Beijing Natural Science Foundation(L182032)。
文摘In this paper,an efficient unequal error protection(UEP)scheme for online fountain codes is proposed.In the buildup phase,the traversing-selection strategy is proposed to select the most important symbols(MIS).Then,in the completion phase,the weighted-selection strategy is applied to provide low overhead.The performance of the proposed scheme is analyzed and compared with the existing UEP online fountain scheme.Simulation results show that in terms of MIS and the least important symbols(LIS),when the bit error ratio is 10-4,the proposed scheme can achieve 85%and 31.58%overhead reduction,respectively.
基金supported by the Natural Science Foundation of Shandong Province,China (Grant No. ZR2021MF049)Joint Fund of Natural Science Foundation of Shandong Province (Grant Nos. ZR2022LLZ012 and ZR2021LLZ001)。
文摘Quantum error correction, a technique that relies on the principle of redundancy to encode logical information into additional qubits to better protect the system from noise, is necessary to design a viable quantum computer. For this new topological stabilizer code-XYZ^(2) code defined on the cellular lattice, it is implemented on a hexagonal lattice of qubits and it encodes the logical qubits with the help of stabilizer measurements of weight six and weight two. However topological stabilizer codes in cellular lattice quantum systems suffer from the detrimental effects of noise due to interaction with the environment. Several decoding approaches have been proposed to address this problem. Here, we propose the use of a state-attention based reinforcement learning decoder to decode XYZ^(2) codes, which enables the decoder to more accurately focus on the information related to the current decoding position, and the error correction accuracy of our reinforcement learning decoder model under the optimisation conditions can reach 83.27% under the depolarizing noise model, and we have measured thresholds of 0.18856 and 0.19043 for XYZ^(2) codes at code spacing of 3–7 and 7–11, respectively. our study provides directions and ideas for applications of decoding schemes combining reinforcement learning attention mechanisms to other topological quantum error-correcting codes.
基金Project supported by Natural Science Foundation of Shandong Province,China (Grant Nos.ZR2021MF049,ZR2022LLZ012,and ZR2021LLZ001)。
文摘Quantum error correction is a crucial technology for realizing quantum computers.These computers achieve faulttolerant quantum computing by detecting and correcting errors using decoding algorithms.Quantum error correction using neural network-based machine learning methods is a promising approach that is adapted to physical systems without the need to build noise models.In this paper,we use a distributed decoding strategy,which effectively alleviates the problem of exponential growth of the training set required for neural networks as the code distance of quantum error-correcting codes increases.Our decoding algorithm is based on renormalization group decoding and recurrent neural network decoder.The recurrent neural network is trained through the ResNet architecture to improve its decoding accuracy.Then we test the decoding performance of our distributed strategy decoder,recurrent neural network decoder,and the classic minimum weight perfect matching(MWPM)decoder for rotated surface codes with different code distances under the circuit noise model,the thresholds of these three decoders are about 0.0052,0.0051,and 0.0049,respectively.Our results demonstrate that the distributed strategy decoder outperforms the other two decoders,achieving approximately a 5%improvement in decoding efficiency compared to the MWPM decoder and approximately a 2%improvement compared to the recurrent neural network decoder.
基金funded by the Key Project of NSFC-Guangdong Province Joint Program(Grant No.U2001204)the National Natural Science Foundation of China(Grant Nos.61873290 and 61972431)+1 种基金the Science and Technology Program of Guangzhou,China(Grant No.202002030470)the Funding Project of Featured Major of Guangzhou Xinhua University(2021TZ002).
文摘Belief propagation list(BPL) decoding for polar codes has attracted more attention due to its inherent parallel nature. However, a large gap still exists with CRC-aided SCL(CA-SCL) decoding.In this work, an improved segmented belief propagation list decoding based on bit flipping(SBPL-BF) is proposed. On the one hand, the proposed algorithm makes use of the cooperative characteristic in BPL decoding such that the codeword is decoded in different BP decoders. Based on this characteristic, the unreliable bits for flipping could be split into multiple subblocks and could be flipped in different decoders simultaneously. On the other hand, a more flexible and effective processing strategy for the priori information of the unfrozen bits that do not need to be flipped is designed to improve the decoding convergence. In addition, this is the first proposal in BPL decoding which jointly optimizes the bit flipping of the information bits and the code bits. In particular, for bit flipping of the code bits, a H-matrix aided bit-flipping algorithm is designed to enhance the accuracy in identifying erroneous code bits. The simulation results show that the proposed algorithm significantly improves the errorcorrection performance of BPL decoding for medium and long codes. It is more than 0.25 d B better than the state-of-the-art BPL decoding at a block error rate(BLER) of 10^(-5), and outperforms CA-SCL decoding in the low signal-to-noise(SNR) region for(1024, 0.5)polar codes.
基金Supported by Headquarters Technology Project of State Grid Corporation of China(No.5700-202118203A-0-0-00)。
文摘Power line communication(PLC)has the potential to become the preferred technique for providing broadband communication to homes and offices with advantage of eliminating the need for new wiring infrastructure and reducing the cost.But it suffers from the impulsive noise because it introduces significant time variance into the power line channel.In this paper,a polar codes based orthogonal frequency division multiplexing(OFDM)PLC system is proposed to deal with the impulsive noise and thereby improve the transmission performance.Firstly,the impulsive noise is modelled with a multi-damped sine function by analyzing the time behavior of impulse events.Then the polar codes are used to combat the impulsive noise of PLC channel,and a low complexity bit-flipping decoding method based on CRC-aided successive cancellation list(CA-SCL)decoding algorithm is proposed.Simulations evaluate the proposed decoding algorithm and the results validate the suggested polar codes based OFDM-PLC scheme which can improve the BER performance of PLC with impulsive interference.
基金supported by the National Key R&D Program of China[Grant Number 2020YFB1708300]the National Natural Science Foundation of China[Grant Number 52075184].
文摘Topology optimization(TO),a numerical technique to find the optimalmaterial layoutwith a given design domain,has attracted interest from researchers in the field of structural optimization in recent years.For beginners,opensource codes are undoubtedly the best alternative to learning TO,which can elaborate the implementation of a method in detail and easily engage more people to employ and extend the method.In this paper,we present a summary of various open-source codes and related literature on TO methods,including solid isotropic material with penalization(SIMP),evolutionary method,level set method(LSM),moving morphable components/voids(MMC/MMV)methods,multiscale topology optimization method,etc.Simultaneously,we classify the codes into five levels,fromeasy to difficult,depending on their difficulty,so that beginners can get started and understand the form of code implementation more quickly.
基金supported in part by the Key Program of National Natural Science Foundation of China (No.92067202)in part by the National Natural Science Foundation of China (No.62071058)in part by the Major Key Project of PCL (PCL2021A15)。
文摘After the pursuit of seventy years,the invention of polar codes indicates that we have found the first capacity-achieving coding with low complexity construction and decoding,which is the great breakthrough of the coding theory in the past two decades.In this survey,we retrospect the history of polar codes and summarize the advancement in the past ten years.First,the primary principle of channel polarization is investigated such that the basic construction,coding method and the classic successive cancellation(SC)decoding are reviewed.Second,in order to improve the performance of the finite code length,we introduce the guiding principle and conclude five design criteria for the construction,design and implementation of the polar code in the practical communication system based on the exemplar schemes in the literature.Especially,we explain the design principle behind the concatenated coding and rate matching of polar codes in 5G wireless system.Furthermore,the improved SC decoding algorithms,such as SC list(SCL)decoding and SC stack(SCS)decoding etc.,are investigated and compared.Finally,the research prospects of polar codes for the future 6G communication system are explored,including the optimization of short polar codes,coding construction in fading channels,polar coded modulation and HARQ,and the polar coded transmission,namely polar processing.Predictably,as a new coding methodology,polar codes will shine a light on communication theory and unveil a revolution in transmission technology.
基金supported in part by the National Key Research and Development Program of China under Grant 2018YFB1802303in part by the Zhejiang Provincial Natural Science Foundation of China under Grant LQ20F010010。
文摘Recently,a generalized successive cancellation list(SCL)decoder implemented with shiftedpruning(SP)scheme,namely the SCL-SP-ωdecoder,is presented for polar codes,which is able to shift the pruning window at mostωtimes during each SCL re-decoding attempt to prevent the correct path from being eliminated.The candidate positions for applying the SP scheme are selected by a shifting metric based on the probability that the elimination occurs.However,the number of exponential/logarithm operations involved in the SCL-SP-ωdecoder grows linearly with the number of information bits and list size,which leads to high computational complexity.In this paper,we present a detailed analysis of the SCL-SP-ωdecoder in terms of the decoding performance and complexity,which unveils that the choice of the shifting metric is essential for improving the decoding performance and reducing the re-decoding attempts simultaneously.Then,we introduce a simplified metric derived from the path metric(PM)domain,and a custom-tailored deep learning(DL)network is further designed to enhance the efficiency of the proposed simplified metric.The proposed metrics are both free of transcendental functions and hence,are more hardware-friendly than the existing metrics.Simulation results show that the proposed DL-aided metric provides the best error correction performance as comparison with the state of the art.
基金supported in part by Joint Fund for Smart Computing of Natural Science Foundation of Shandong Province(ZR2019LZH001)Shandong University Youth Innovation Supporting Program(2019KJN020,2019KJN024)+1 种基金Shandong Key Research and Development Project(2019GGX101066)the Taishan Scholar Program of Shandong Province,the Natural Science Foundation of China(61701284).
文摘To remove the restriction on code length of polar codes,this paper proposes a construction scheme,called stepwise polar codes,which can gen-erate arbitrary-length polar codes.The stepwise polar codes are generated by sub-polar codes with different code lengths.To improve coding performance,sub-polar codes are united by polarization effect priority algorithm,which can reduce the number of in-completely polarized channels.Then,the construction method of the generator matrix of the stepwise po-lar code is presented.Furthermore,we prove that the proposed scheme has lower decoding complexity than punctured,multi-kernel polar codes.Simulation results show that the proposed method can achieve similar decoding performance compared with the conventional punctured polar codes,rate-compatible punctured polar code,PC-short and asymmetric polar codes(APC)when code length N=48 and 72,respectively.
文摘Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.
基金supported in part with the Project on the Industry Key Technologies of Jiangsu Province(No.BE2017153)the Industry-University-Research Fund of ZTE Corporation.
文摘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 protograph 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.
基金the National Natural Science Foundation of China (Grant Nos. 61972413, 61901525, and 62002385)the National Key R&D Program of China (Grant No. 2021YFB3100100)RGC under Grant No. N HKUST619/17 from Hong Kong, China。
文摘In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the Mac Williams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.
基金supported by the Fundamental Research Funds for the Central Universities (FRF-TP20-062A1)Guangdong Basic and Applied Basic Research Foundation (2021A1515110070)
文摘Belief propagation(BP)decoding outputs soft information and can be naturally used in iterative receivers.BP list(BPL)decoding provides comparable error-correction performance to the successive cancellation list(SCL)decoding.In this paper,we firstly introduce an enhanced code construction scheme for BPL decoding to improve its errorcorrection capability.Then,a GPU-based BPL decoder with adoption of the new code construction is presented.Finally,the proposed BPL decoder is tested on NVIDIA RTX3070 and GTX1060.Experimental results show that the presented BPL decoder with early termination criterion achieves above 1 Gbps throughput on RTX3070 for the code(1024,512)with 32 lists under good channel conditions.
文摘As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.
基金This work is partially supported by the National Key Research and Development Project under Grant 2018YFB1802402.
文摘The error correction performance of Belief Propagation(BP)decoding for polar codes is satisfactory compared with the Successive Cancellation(SC)decoding.Nevertheless,it has to complete a fixed number of iterations,which results in high computational complexity.This necessitates an intelligent identification of successful BP decoding for early termination of the decoding process to avoid unnecessary iterations and minimize the computational complexity of BP decoding.This paper proposes a hybrid technique that combines the“paritycheck”with the“G-matrix”to reduce the computational complexity of BP decoder for polar codes.The proposed hybrid technique takes advantage of the parity-check to intelligently identify the valid codeword at an early stage and terminate the BP decoding process,which minimizes the overhead of the G-matrix and reduces the computational complexity of BP decoding.We explore a detailed mechanism incorporating the parity bits as outer code and prove that the proposed hybrid technique minimizes the computational complexity while preserving the BP error correction performance.Moreover,mathematical formulation for the proposed hybrid technique that minimizes the computation cost of the G-matrix is elaborated.The performance of the proposed hybrid technique is validated by comparing it with the state-of-the-art early stopping criteria for BP decoding.Simulation results show that the proposed hybrid technique reduces the iterations by about 90%of BP decoding in a high Signal-to-Noise Ratio(SNR)(i.e.,3.5~4 dB),and approaches the error correction performance of G-matrix and conventional BP decoder for polar codes.
基金supported in part by the National Natural Science Foundation of China(No.62071026,No.62201152 and No.61941106)the Natural Science Foundation of Fujian Province(No.2021J05034)Key Project of Science and Technology Innovation of Fujian Province(No.2021G02006)。
文摘Low-density parity-check(LDPC)codes are not only capacity-approaching,but also greatly suitable for high-throughput implementation.Thus,they are the most popular codes for high-speed data transmission in the past two decades.Thanks to the low-density property of their parity-check matrices,the optimal maximum a posteriori probability decoding of LDPC codes can be approximated by message-passing decoding with linear complexity and highly parallel nature.Then,it reveals that the approximation has to carry on Tanner graphs without short cycles and small trapping sets.Last,it demonstrates that well-designed LDPC codes with the aid of computer simulation and asymptotic analysis tools are able to approach the channel capacity.Moreover,quasi-cyclic(QC)structure is introduced to significantly facilitate their high-throughput implementation.In fact,compared to the other capacity-approaching codes,QC-LDPC codes can provide better area-efficiency and energy-efficiency.As a result,they are widely applied in numerous communication systems,e.g.,Landsat satellites,Chang’e Chinese Lunar mission,5G mobile communications and so on.What’s more,its extension to non-binary Galois fields has been adopted as the channel coding scheme for BeiDou navigation satellite system.
基金supported in part by the National Key R&D Program of China(No.2019YFB1803400)。
文摘For polar codes,the performance of successive cancellation list(SCL)decoding is capable of approaching that of maximum likelihood decoding.However,the existing hardware architectures for the SCL decoding suffer from high hardware complexity due to calculating L decoding paths simultaneously,which are unfriendly to the devices with limited logical resources,such as field programmable gate arrays(FPGAs).In this paper,we propose a list-serial pipelined hardware architecture with low complexity for the SCL decoding,where the serial calculation and the pipelined operation are elegantly combined to strike a balance between the complexity and the latency.Moreover,we employ only one successive cancellation(SC)decoder core without L×L crossbars,and reduce the number of inputs of the metric sorter from 2L to L+2.Finally,the FPGA implementations show that the hardware resource consumption is significantly reduced with negligible decoding performance loss.
基金Project supported by the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2021MF049,ZR2022LLZ012,and ZR2021LLZ001)。
文摘Quantum error correction technology is an important method to eliminate errors during the operation of quantum computers.In order to solve the problem of influence of errors on physical qubits,we propose an approximate error correction scheme that performs dimension mapping operations on surface codes.This error correction scheme utilizes the topological properties of error correction codes to map the surface code dimension to three dimensions.Compared to previous error correction schemes,the present three-dimensional surface code exhibits good scalability due to its higher redundancy and more efficient error correction capabilities.By reducing the number of ancilla qubits required for error correction,this approach achieves savings in measurement space and reduces resource consumption costs.In order to improve the decoding efficiency and solve the problem of the correlation between the surface code stabilizer and the 3D space after dimension mapping,we employ a reinforcement learning(RL)decoder based on deep Q-learning,which enables faster identification of the optimal syndrome and achieves better thresholds through conditional optimization.Compared to the minimum weight perfect matching decoding,the threshold of the RL trained model reaches 0.78%,which is 56%higher and enables large-scale fault-tolerant quantum computation.
基金supported in part by the National Natural Science Foundation of China under Grant 92067202,Grant 62071058.
文摘In this paper,we propose an arbitrary decode-forward single-relay scheme for finite blocklength polar codes,which can be applied to the general symmetric discrete memoryless relay channel with orthogonal receiver components.The relay node decodes the received message.The relay node selectively re-encodes the message and transmits it to the destination node.Furthermore,in order to minimize the upper-bound of the block error probability,we propose a selection strategy to decide the proper re-encoded bit set by the relay.Simulation results are presented to illustrate the improvement in decoding performance of the proposed scheme compared to conventional relay schemes in both additive white Gaussian noise(AWGN)channel and Rayleigh fading channel(RFC).