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Quantum secret sharing based on quantum error-correcting codes 认领 引用 被引量：3
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作者 张祖荣 刘伟涛 李承祖 《Chinese Physics B》 SCIE EI CAS CSCD 2011年第5期91-95,共5页
Quantum secret sharing（QSS） is a procedure of sharing classical information or quantum information by using quantum states. This paper presents how to use a [2k- 1, 1, k] quantum error-correcting code （QECC） to im... Quantum secret sharing（QSS） is a procedure of sharing classical information or quantum information by using quantum states. This paper presents how to use a [2k- 1, 1, k] quantum error-correcting code （QECC） to implement a quantum （k, 2k-1） threshold scheme. It also takes advantage of classical enhancement of the [2k-1, 1, k] QECC to establish a QSS scheme which can share classical information and quantum information simultaneously. Because information is encoded into QECC, these schemes can prevent intercept-resend attacks and be implemented on some noisy channels. 展开更多
关键词 quantum secret sharing quantum error-correcting code classically enhanced quantumerror-correcting code
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Quantum quasi-cyclic low-density parity-check error-correcting codes 认领 引用 被引量：5
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作者 李渊 曾贵华 Moon Ho Leeb 《Chinese Physics B》 SCIE EI CAS 2009年第10期4154-4160,共7页
In this paper, we propose the approach of employing circulant permutation matrices to construct quantum quasicyclic （QC） low-density parity-check （LDPC） codes. Using the proposed approach one may construct some ne... In this paper, we propose the approach of employing circulant permutation matrices to construct quantum quasicyclic （QC） low-density parity-check （LDPC） codes. Using the proposed approach one may construct some new quantum codes with various lengths and rates of no cycles-length 4 in their Tanner graphs. In addition, these constructed codes have the advantages of simple implementation and low-complexity encoding. Finally, the decoding approach for the proposed quantum QC LDPC is investigated. 展开更多
关键词 quantum LDPC code quasi-cycliC circulant permutation matrix CSS code
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On the complete weight distributions of quantum error-correcting codes 认领 引用 被引量：1
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作者 杜超 马智 熊茂胜 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第5期272-287,共16页
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
关键词 quantum codes complete weight distributions Mac Williams identities BCH codes
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Study on the aided PC-ASCL decoding algorithm based on optimized parity check codes 认领 引用
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作者 Jianguo YUAN Deyi YANG +1 位作者 Ziliang ZHU Jinneng XIANG 《Optoelectronics Letters》 EI 2026年第2期87-91,共5页
Aiming at the poor performance of the parity check(PC) aided adaptive successive cancellation list(PC-ASCL) decoding algorithm because the PC code in the polar code can only verify odd errors, an optimized parity chec... Aiming at the poor performance of the parity check(PC) aided adaptive successive cancellation list(PC-ASCL) decoding algorithm because the PC code in the polar code can only verify odd errors, an optimized parity check(OPC) code which can verify all odd errors as well as the half even errors is proposed. The OPC code is used to improve the PC-ASCL decoding algorithm, thus an OPC aided ASCL(OPC-ASCL) decoding algorithm is proposed. In the coding stage, the algorithm divides the information sequence into multiple segments, and places an OPC code at the end of each segment to verify the current information sequence, and places a cyclic redundancy check code at the end of the entire information sequence to verify the entire information sequence. In the decoding stage, the algorithm uses the OPC-ASCL decoder to decode. Simulation results show that compared to the PC-ASCL decoding algorithm, the OPC-ASCL decoding algorithm can reduce the complexity and obtain the certain performance gain. 展开更多
关键词 pc ascl decoding pc code adaptive successive cancellation list pc ascl divides information sequence multip parity check polar code optimized parity check coding stage
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Methods,Algorithms,Codes,and Software for Theoretical and Computational Chemistry 认领 引用
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作者 Bin Jiang 《Chinese Journal of Chemical Physics》 SCIE EI CAS CSCD 2026年第3期301-302,共2页
Theoretical and computational chemistry has profoundly impacted a wide range of disciplines,from chemistry and physics to biology and materials science.In recent years,remarkable advances in electronic structure theor... Theoretical and computational chemistry has profoundly impacted a wide range of disciplines,from chemistry and physics to biology and materials science.In recent years,remarkable advances in electronic structure theory,molecular dynamics,and machine learning methods——coupled with increasingly powerful algorithms and software—have equipped chemists with an unprecedented arsenal of tools to tackle complex chemical problems. 展开更多
关键词 theoretical computational chemistry electronic structure theorymolecular dynamicsand algorithms codes software methods materials sciencein theoretical chemistry
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Open-set blind recognition of non-binary LDPC codes 认领 引用 被引量：1
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作者 Ming Jiang Dongrui Huan +1 位作者 Shiqiu Liu Junhui Zhao 《Digital Communications and Networks》 SCIE EI CSCD 2026年第5期836-844,共9页
Blind encoder recognition methods have drawn much research interest in recent years,as they can play an important role in non-cooperative scenarios.This paper proposes an open-set blind recognition method for nonbinar... Blind encoder recognition methods have drawn much research interest in recent years,as they can play an important role in non-cooperative scenarios.This paper proposes an open-set blind recognition method for nonbinary Low-Density Parity-Check(LDPC)codes with high performance.We derive the principle of obtaining a non-binary Parity-Check Matrix(PCM)from the received signal and further apply the Gauss-Jordan Column Elimination(GJCE)over GF(q).An iterative process involving GJCE,LDPC decoding,and PCM sparsification is applied in our proposed method.The proposed symbol-level blind recognition is shown to perform much better than the bit-level one.Experimental results demonstrate that our proposed method can completely reconstruct the PCMs at a bit error rate of 0.4%for the moderate-length non-binary LDPC codes. 展开更多
关键词 Blind recognition Symbol-level Open-set Non-binary LDPC codes
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Rateless Polar Codes with Unequal Error Protection Property 认领 引用
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作者 Cui Chen Xiang Wei +1 位作者 Ma Siwei Guo Qing 《China Communications》 SCIE EI CSCD 2026年第1期10-23,共14页
Mobile communications are reaching out to every aspect of our daily life,necessitating highefficiency data transmission and support for diverse data types and communication scenarios.Polar codes have emerged as a prom... Mobile communications are reaching out to every aspect of our daily life,necessitating highefficiency data transmission and support for diverse data types and communication scenarios.Polar codes have emerged as a promising solution due to their outstanding error-correction performance and low complexity.Unequal error protection(UEP)involves nonuniform error safeguarding for distinct data segments,achieving a fine balance between error resilience and resource allocation,which ultimately enhancing system performance and efficiency.In this paper,we propose a novel class of UEP rateless polar codes.The codes are designed based on matrix extension of polar codes,and elegant mapping and duplication operations are designed to achieve UEP property while preserving the overall performance of conventional polar codes.Superior UEP performance is attained without significant modifications to conventional polar codes,making it straightforward for compatibility with existing polar codes.A theoretical analysis is conducted on the block error rate and throughput efficiency performance.To the best of our knowledge,this work provides the first theoretical performance analysis of UEP rateless polar codes.Simulation results show that the proposed codes significantly outperform existing polar coding schemes in both block error rate and throughput efficiency. 展开更多
关键词 matrix extension polar codes rateless coding unequal error protection
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New binary quantum stabilizer codes from classical quasi-cyclic codes of index two 认领 引用
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作者 Zhen-Hui Zhang Zhuo Li Li-Juan Xing 《Chinese Physics B》 SCIE EI CAS CSCD 2026年第4期253-264,共12页
We study the algebraic structures of a generalized class of quasi-cyclic codes of index two,and prove that any quasicyclic code of index two can be transformed into our proposed quasi-cyclic codes.We determine the gen... We study the algebraic structures of a generalized class of quasi-cyclic codes of index two,and prove that any quasicyclic code of index two can be transformed into our proposed quasi-cyclic codes.We determine the generator forms of the proposed quasi-cyclic codes and their Hermitian dual codes.We establish the necessary and sufficient conditions for Hermitian self-orthogonality and dual-containing property to construct quantum stabilizer codes via the Hermitian construction method.As an application,10 record-breaking quantum stabilizer codes are constructed. 展开更多
关键词 quasi-cyclic codes quantum stabilizer codes Hermitian inner product Hermitian construction
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Efficient Polar Codes with Low Complexity for Correcting Insertions/Deletions in DPPM 认领 引用
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作者 Li Leran Liu Yuan +2 位作者 Yuan Ye Xiahou Wenqian Chen Maonan 《China Communications》 SCIE EI CSCD 2026年第1期24-33,共10页
Differential pulse-position modulation(DP PM)can achieve a good compromise between power and bandwidth requirements.However,the output sequence has undetectable insertions and deletions.This paper proposes a successiv... Differential pulse-position modulation(DP PM)can achieve a good compromise between power and bandwidth requirements.However,the output sequence has undetectable insertions and deletions.This paper proposes a successive cancellation(SC)decoding scheme based on the weighted levenshtein distance(WLD)of polar codes for correcting insertions/deletions in DPPM systems.In this method,the WLD is used to calculate the transfer probabilities recursively to obtain likelihood ratios,and the low-complexity SC decoding method is built according to the error characteristics to match the DPPM system.Additionally,the proposed SC decoding scheme is extended to list decoding,which can further improve error correction performance.Simulation results show that the proposed scheme can effectively correct insertions/deletions in the DPPM system,which enhances its reliability and performance. 展开更多
关键词 DPPM insertions/deletions polar codes SC decoding
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Two Bit-Serial Architectures for Check Node Update in Min-Sum Decoding of LDPC Codes 认领 引用
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作者 Hu Qing He Xuan +2 位作者 Luo Lisha Cai Kui Tang Xiaohu 《China Communications》 SCIE EI CSCD 2026年第3期142-150,共9页
Implementing check node(CN)update based on the minimum value(MV)and second MV of incoming message magnitudes is crucial for Min-Sum Algorithms(MSAs).In the category of bit-serial implementations,existing schemes suffe... Implementing check node(CN)update based on the minimum value(MV)and second MV of incoming message magnitudes is crucial for Min-Sum Algorithms(MSAs).In the category of bit-serial implementations,existing schemes suffer from decoding performance degradation,large hardware areas,and/or long latency.In this paper,we propose two efficient CN update functions based on the MV and an approximate second MV,and design bit-serial architectures to implement them.Simulation results show that our functions exhibit the minimum decoding performance degradation compared to the existing functions using approximate second MVs.Moreover,the applicationspecific integrated circuits(ASIC)implementation results demonstrate the advantages of our architectures in terms of area,latency,etc. 展开更多
关键词 bit-serial check node update hardware implementation LDPC code min-sum algorithm
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Multi-Attempt List Decoding of Polar Codes 认领 引用
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作者 Yuan Peihong Chen Zhe +1 位作者 Wu Yongpeng Gao Yue 《China Communications》 SCIE EI CSCD 2026年第4期27-36,共10页
In this work,we propose a multi-attempt successive cancellation list(MA-SCL)decoder for polar codes that achieves identical error-correction performance to standard SCL decoding while reducing average complexity.Unlik... In this work,we propose a multi-attempt successive cancellation list(MA-SCL)decoder for polar codes that achieves identical error-correction performance to standard SCL decoding while reducing average complexity.Unlike CRC-aided SCL,the proposed MA-SCL progressively restarts decoding with increasing list sizes and reuses information from previous attempts.This design eliminates the need for outer CRC codes.The decoder features dynamic searchspace pruning and an early stopping criterion based on path metrics.Simulations show MA-SCL matches SCL performance with lower average complexity,particularly for short polar-like codes with reed-muller(RM)rate profiles and dynamic frozen constraints.Compared to existing adaptive decoders,MA-SCL offers implementation advantages by eliminating the need for stack-/heap management while providing relatively stable latency bounds(1×to|Λ|×SCL latency). 展开更多
关键词 complexity-adaptive decoding list decoding polar coding
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Design of Interleaved PAC Codes for Meteor Burst Channels 认领 引用
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作者 Wu Yunzhi Li Li Tang Xiaohu 《China Communications》 SCIE EI CSCD 2026年第5期31-43,共13页
Meteor burst channel demonstrates special fading characteristics,in which the statistics of channel envelope along the time domain explicitly violate the traditional iid(independent and identically distributed)assumpt... Meteor burst channel demonstrates special fading characteristics,in which the statistics of channel envelope along the time domain explicitly violate the traditional iid(independent and identically distributed)assumption.Because this kind of channel will rapidly disappear after its occurrence.In order to adapt to this particular characteristic of meteor burst channel,a noniid assumption based Gaussian approximation(GA)algorithm is employed to rebuild the state-of-the-art polarization adjusted convolutional(PAC)code.Then,the polarization effect under non-iid condition is analyzed and an interleaving method is employed to enhance the polarization efficiency.Compared with the widely used low density parity check(LDPC)codes and turbo product codes(TPC),our interleaved PAC codes are capable of decreasing the BLER level by one order of magnitude. 展开更多
关键词 interleaving meteor burst channel PAC codes
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On Some Dually-BCH Codes 认领 引用
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作者 Pu Keli Liao Qunying 《Journal of Mathematical Research with Applications》 CSCD 2026年第4期465-480,共16页
Bose-Chaudhuri-Hocquenghem(BCH)codes have wide applications in communication and storage.In this article,for given BCH code C(q,n,δ,b)(b=0,1)with respect to β of length n=qm+1,we present several sufficient con... Bose-Chaudhuri-Hocquenghem(BCH)codes have wide applications in communication and storage.In this article,for given BCH code C(q,n,δ,b)(b=0,1)with respect to β of length n=qm+1,we present several sufficient conditions for C(q,n,δ,b)(b=0,1)to be dually-BCH or not,respectively.Especially,for some designed distance δ,we determine the parameters of the narrow-sense BCH code C(q,q2+1,δ,1),and for q=2l with l≥2,we provide a sufficient condition for C(q,q2+1,δ,0) dually-BCH and determine the parameters of both C(q,q2+1,δ,0) and the dual. 展开更多
关键词 BCH code dual code dually-BCH code cyclotomic coset
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Adaptive Coding and Modulation Using Enhanced Turbo Codes for Meteor Burst Communication 认领 引用
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作者 Ding Mengchuan Li Zhiyong +1 位作者 Wang Wei Li Xuekun 《China Communications》 SCIE EI CSCD 2026年第5期1-14,共14页
Meteor burst channels(MBC)exhibit significant randomness and non-stationarity,limiting the effectiveness of traditional fixed-rate transmission strategies.This paper proposes an adaptive coding and modulation(ACM)sche... Meteor burst channels(MBC)exhibit significant randomness and non-stationarity,limiting the effectiveness of traditional fixed-rate transmission strategies.This paper proposes an adaptive coding and modulation(ACM)scheme based on real-time channel state perception using enhanced Turbo codes.By leveraging the interleaving structure of Turbo codes to mitigate channel fading and incorporating multi-rate adaptation,the scheme dynamically adjusts symbol rate,modulation,and coding methods to improve spectral efficiency.Meanwhile,a joint evaluation mechanism integrating physical-layer signal to noise ratio(SNR)estimation and media access control(MAC)-layer frame error rate(FER)statistics is introduced to achieve dynamic optimization of switching thresholds in multipath channels.Simulation and experimental results demonstrate that the proposed scheme significantly enhances system adaptability to MBC,and increases data transmission success rates,offering a more efficient and reliable solution for MBC systems. 展开更多
关键词 ACM enhanced turbo codes MAC-layer meteor burst channels
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TLS Blind Recognition Algorithm of LDPC Codes 认领 引用
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作者 Ning Xiaoyan Sun Jingjing +1 位作者 Wang Zhenduo Sun Zhiguo 《China Communications》 SCIE EI CSCD 2026年第2期112-121,共10页
Blind recognition of low-density paritycheck(LDPC)codes has gradually attracted more attention with the development of military and civil communications.However,in the case of the paritycheck matrices with relatively ... Blind recognition of low-density paritycheck(LDPC)codes has gradually attracted more attention with the development of military and civil communications.However,in the case of the paritycheck matrices with relatively high row weights,the existing blind recognition algorithms based on a candidate set generally perform worse.In this paper,we propose a blind recognition method for LDPC codes,called as tangent function assisted least square(TLS)method,which improves recognition performances by constructing a new cost function.To characterize the constraint degree among received vectors and paritycheck vectors,a feature function based on tangent function is constructed in the proposed algorithm.A cost function based on least square method is also established according to the feature function values satisfying the parity-check relationship.Moreover,the minimum average value in TLS is obtained on the candidate set.Numerical analysis and simulation results show that recognition performances of TLS algorithm are consistent with theoretical results.Compared with existing algorithms,the proposed method possesses better recognition performances. 展开更多
关键词 blind recognition cost function least square low-density parity-check(LDPC)codes
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Cryptanalysis on AW digital signature scheme based on error-correcting codes 认领 引用 被引量：3
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作者 张振峰 冯登国 戴宗铎 《Science in China(Series F)》 2002年第5期397-400,共4页
In 1993, Alabhadi and Wicker gave a modification to Xinmei Digital Signature Scheme based on error-correcting codes, which is usually denoted by AW Scheme. In this paper we show that the AW Scheme is actually not secu... In 1993, Alabhadi and Wicker gave a modification to Xinmei Digital Signature Scheme based on error-correcting codes, which is usually denoted by AW Scheme. In this paper we show that the AW Scheme is actually not secure: anyone holding public keys of the signatory can obtain the equivalent private keys, and then forge digital signatures for arbitrary messages successfully. We also point out that one can hardly construct a digital signature scheme with high-level security due to the difficulty of decomposing large matrixes. 展开更多
关键词 cryptanalysis digital signature error-correcting code.
An Error-Correcting Code-Based Robust Watermarking Scheme for Stereolithographic Files 认领 引用 被引量：2
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作者 Zhuorong Li Huawei Tian +2 位作者 Yanhui Xiao Yunqi Tang Anhong Wang 《Computer Systems Science & Engineering》 SCIE EI 2021年第5期247-263,共17页
Stereolithographic(STL)files have been extensively used in rapid prototyping industries as well as many other fields as watermarking algorithms to secure intellectual property and protect three-dimensional models from... Stereolithographic(STL)files have been extensively used in rapid prototyping industries as well as many other fields as watermarking algorithms to secure intellectual property and protect three-dimensional models from theft.However,to the best of our knowledge,few studies have looked at how watermarking can resist attacks that involve vertex-reordering.Here,we present a lossless and robust watermarking scheme for STL files to protect against vertexreordering attacks.Specifically,we designed a novel error-correcting code(ECC)that can correct the error of any one-bit in a bitstream by inserting several check digits.In addition,ECC is designed to make use of redundant information according to the characteristics of STL files,which introduces further robustness for defense against attacks.No modifications are made to the geometric information of the three-dimensional model,which respects the requirements of a highprecision model.The experimental results show that the proposed watermarking scheme can survive numerous kinds of attack,including rotation,scaling and translation(RST),facet reordering,and vertex-reordering attacks. 展开更多
关键词 Three-dimensional watermarking stereolithographic file robust watermarking error-correcting code
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Reliability Performance Estimation and Its Applications of Rate-Compatible Polar Codes for B5G-IoT 认领 引用
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作者 Liang Hao Liang Xiaohu +2 位作者 Ye Ganhua Lu Ruimin Lu Xinjin 《China Communications》 SCIE EI CSCD 2025年第7期124-137,共14页
The beyond fifth-generation Internet of Things requires more capable channel coding schemes to achieve high-reliability,low-complexity and lowlatency communications.The theoretical analysis of error-correction perform... The beyond fifth-generation Internet of Things requires more capable channel coding schemes to achieve high-reliability,low-complexity and lowlatency communications.The theoretical analysis of error-correction performance of channel coding functions as a significant way of optimizing the transmission reliability and efficiency.In this paper,the efficient estimation methods of the block error rate(BLER)performance for rate-compatible polar codes(RCPC)are proposed under several scenarios.Firstly,the BLER performance of RCPC is generally evaluated in the additive white Gaussian noise channels.That is further extended into the Rayleigh fading channel case using an equivalent estimation method.Moreover,with respect to the powerful decoder such as successive cancellation list decoding,the performance estimation is derived analytically based on the polar weight spectrum and BLER upper bounds.Theoretical evaluation and numerical simulation results show that the estimated performance can fit well the practical simulated results of RCPC under the objective conditions,verifying the validity of our proposed performance estimation methods.Furthermore,the application designs of the reliability estimation of RCPC are explored,particularly in the advantages of the signal-to-noise(SNR)estimation and throughput efficiency optimization of polar coded hybrid automatic repeat request. 展开更多
关键词 Internet of Things polar codes ratecompatible reliability estimation SNR estimation throughput optimization
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Computation-Efficient Decoding of LDPC Codes for High-Speed Space Laser Communications 认领 引用
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作者 Hu Zhuojun Chen Zhao +1 位作者 Kuang Linling Yin Liuguo 《China Communications》 SCIE EI CSCD 2025年第12期108-123,共16页
Space laser communication(SLC)is an emerging technology to support high-throughput data transmissions in space networks.In this paper,to guarantee the reliability of high-speed SLC links,we aim at practical implementa... Space laser communication(SLC)is an emerging technology to support high-throughput data transmissions in space networks.In this paper,to guarantee the reliability of high-speed SLC links,we aim at practical implementation of low-density paritycheck(LDPC)decoding under resource-restricted space platforms.Particularly,due to the supply restriction and cost issues of high-speed on-board devices such as analog-to-digital converters(ADCs),the input of LDPC decoding will be usually constrained by hard-decision channel output.To tackle this challenge,density-evolution-based theoretical analysis is firstly performed to identify the cause of performance degradation in the conventional binaryinitialized iterative decoding(BIID)algorithm.Then,a computation-efficient decoding algorithm named multiary-initialized iterative decoding with early termination(MIID-ET)is proposed,which improves the error-correcting performance and computation efficiency by using a reliability-based initialization method and a threshold-based decoding termination rule.Finally,numerical simulations are conducted on example codes of rates 7/8 and 1/2 to evaluate the performance of different LDPC decoding algorithms,where the proposed MIID-ET outperforms the BIID with a coding gain of 0.38 dB and variable node calculation saving of 37%.With this advantage,the proposed MIID-ET can notably reduce LDPC decoder’s hardware implementation complexity under the same bit error rate performance,which successfully doubles the total throughput to 10 Gbps on a single-chip FPGA. 展开更多
关键词 computation-efficient decoding highspeed decoders LDPC codes LLR initialization space laser communications
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Low Complexity Successive Cancellation List Decoding of U-UV Codes 认领 引用
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作者 Chen Wenhao Chen Li +1 位作者 Lin Jingyu Zhang Huazi 《China Communications》 SCIE EI CSCD 2025年第1期41-60,共20页
Constituted by BCH component codes and its ordered statistics decoding(OSD),the successive cancellation list(SCL)decoding of U-UV structural codes can provide competent error-correction performance in the short-to-med... Constituted by BCH component codes and its ordered statistics decoding(OSD),the successive cancellation list(SCL)decoding of U-UV structural codes can provide competent error-correction performance in the short-to-medium length regime.However,this list decoding complexity becomes formidable as the decoding output list size increases.This is primarily incurred by the OSD.Addressing this challenge,this paper proposes the low complexity SCL decoding through reducing the complexity of component code decoding,and pruning the redundant SCL decoding paths.For the former,an efficient skipping rule is introduced for the OSD so that the higher order decoding can be skipped when they are not possible to provide a more likely codeword candidate.It is further extended to the OSD variant,the box-andmatch algorithm(BMA),in facilitating the component code decoding.Moreover,through estimating the correlation distance lower bounds(CDLBs)of the component code decoding outputs,a path pruning(PP)-SCL decoding is proposed to further facilitate the decoding of U-UV codes.In particular,its integration with the improved OSD and BMA is discussed.Simulation results show that significant complexity reduction can be achieved.Consequently,the U-UV codes can outperform the cyclic redundancy check(CRC)-polar codes with a similar decoding complexity. 展开更多
关键词 ordered statistics decoding successive cancellation list decoding U-UV codes
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