In this paper, the maximal length of maximal distance separable (MDS) codes is studied, and a new upper bound formula of the maximal length of MDS codes is obtained. Especially, the exact values of the maximal length ...In this paper, the maximal length of maximal distance separable (MDS) codes is studied, and a new upper bound formula of the maximal length of MDS codes is obtained. Especially, the exact values of the maximal length of MDS codes in some parameters are given.展开更多
Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of...Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.展开更多
In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe ...In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe Chapote, these maximal codes are constructed over finite fields, and these codes are used for coding and decoding.展开更多
Optical orthogonal code is the main signature code employed by optical CDMA system. Starting from modern mathematics theory, finite projective geometry and Galois theory, the essential connection between optical ortho...Optical orthogonal code is the main signature code employed by optical CDMA system. Starting from modern mathematics theory, finite projective geometry and Galois theory, the essential connection between optical orthogonal code designing and finite geometry theory were discussed; find out the corresponding relationship between the parameter of OOC and that of finite geometry space. In this article, the systematic theory of OOC designing based on projective geometry is established in detail. The designing process and results of OOC on projective plane PG(2,q) and on m-dimension projective space are given respectively. Furthermore, the analytical theory for the corresponding relation between OOC with high cross-correlation and k-D manifold of projective space is set up. The OOC designing results given in this article have excellent performance, whose maximum cross-correlation is 1, and the cardinality reaches the Johnson upper bound, i.e. it realizes the optimization in both MUI and system capacity.展开更多
Let E be an Archimedean Riesz algebra possessing a weak unit element e and a maximal disjoint system {e,: i∈I} in which e, is a projection element for each i. The principal band generated by eiis denoted by B(ei). T...Let E be an Archimedean Riesz algebra possessing a weak unit element e and a maximal disjoint system {e,: i∈I} in which e, is a projection element for each i. The principal band generated by eiis denoted by B(ei). The main result in this paper says that if there exists a completely regular Hausdorff space X such that E is Riesz algebra isomorphic to C(X) then for every i ∈ I there exists a completely regular Hausdorff space X, such that B(ei) is Riesz algebra isomorphic to C(Xi). Under an additional condition the inverse holds.展开更多
The codes of formal power series rings R_∞=F[[r]]={sum from i=0 to ∞(a_lr^l|a_l∈F)}and finite chain rings R_i={a_0+a_1r+…+a_(i-1)r^(i-1)|a_i∈F}have close relationship in lifts and projection.In this paper,we stud...The codes of formal power series rings R_∞=F[[r]]={sum from i=0 to ∞(a_lr^l|a_l∈F)}and finite chain rings R_i={a_0+a_1r+…+a_(i-1)r^(i-1)|a_i∈F}have close relationship in lifts and projection.In this paper,we study self-dual codes over R_∞by means of self-dual codes over Ri,and give some characterizations of self-dual codes over R_∞.展开更多
In this paper,we exhibit a free monoid containing all prefix codes in connection with the sets of i-th powers of primitive words for all i≥2.This extends two results given by Shyr and Tsai in 1998 at the same time.
In this paper,we innovatively associate the mutual information with the frame error rate(FER)performance and propose novel quantized decoders for polar codes.Based on the optimal quantizer of binary-input discrete mem...In this paper,we innovatively associate the mutual information with the frame error rate(FER)performance and propose novel quantized decoders for polar codes.Based on the optimal quantizer of binary-input discrete memoryless channels(BDMCs),the proposed decoders quantize the virtual subchannels of polar codes to maximize mutual information(MMI)between source bits and quantized symbols.The nested structure of polar codes ensures that the MMI quantization can be implemented stage by stage.Simulation results show that the proposed MMI decoders with 4 quantization bits outperform the existing nonuniform quantized decoders that minimize mean-squared error(MMSE)with 4 quantization bits,and yield even better performance than uniform MMI quantized decoders with 5 quantization bits.Furthermore,the proposed 5-bit quantized MMI decoders approach the floating-point decoders with negligible performance loss.展开更多
Random Projection Code (RPC) is a mechanism that combines channel coding and modulation together and realizes rate adaptation in the receiving end. Random projection code’s mapping matrix has significant influences o...Random Projection Code (RPC) is a mechanism that combines channel coding and modulation together and realizes rate adaptation in the receiving end. Random projection code’s mapping matrix has significant influences on decoding performance as well as hardware implementation complexity. To reduce hardware implementation complexity, we design a quasi-cyclic mapping matrix for RPC codes. Compared with other construction approaches, our design gets rid of data filter component, thus reducing chip area 7284.95 um2, and power consumption 331.46 uW in 0.13 um fabrication. Our simulation results show that our method does not cause any performance loss and even gets 0.2 dB to 0.5 dB gain at BER 10-4.展开更多
The authors p oint out a problem in the article of Ref.(Xiong Hongyun,Rong Ximin.Maximal disjoint systems in Riesz space and representation.Acta Math Sinica ,1998,41(4):763-766.)and revise it.Let E be an Archimed ean ...The authors p oint out a problem in the article of Ref.(Xiong Hongyun,Rong Ximin.Maximal disjoint systems in Riesz space and representation.Acta Math Sinica ,1998,41(4):763-766.)and revise it.Let E be an Archimed ean Riesz space possessing a weak unit e and a maximal disjoint division{e i:i∈I} in which each e i is a proj ection element. Concerning the following statements:(1) There exists a completel y regular Hausdorff space X such that E is Riesz isomorphic to C(X);(2) For every i∈I there exi sts a completely regular Hausdorff space X i such that the band generated by e i is Riesz isomorphic to C(X i).It is shown that (1) implies (2),and find some conditions for the inverse bein g true are found.Furthermore,if each X i in (2) is a compact Ha usdorff space, a necessary and sufficient condition is established under which E can be represented as a C(X) for some compact Hausdorff space X.As corollaries, corresponding results for late rally complete Riesz spaces are obtained.展开更多
The most popular and representative classic waveform codes are referred to as orthogonal,bi-orthogonal,simplex,and etc,but the choice of waveform codes is essentially identical in error performance and cross correlati...The most popular and representative classic waveform codes are referred to as orthogonal,bi-orthogonal,simplex,and etc,but the choice of waveform codes is essentially identical in error performance and cross correlation characteristic.Though bi-orthogonal coding requires half the bandwidth of the others,such coding scheme is attractive only when large bandwidth is available.In this paper,a novel finite projective plane（FPP） based waveform coding scheme is proposed,which is with similar error performance and cross correlation.Nevertheless,the bandwidth requirement will grow in a quadratic way,but not in an exponential way with the values of message bit numbers（k）.The proposed scheme takes obvious advantages over the bi-orthogonal scheme when k ≥ 6.展开更多
In this paper, we propose to generalize the coding schemes first proposed by Kozic &al to high spectral efficient modulation schemes. We study at first Chaos Coded Modulation based on the use of small ...In this paper, we propose to generalize the coding schemes first proposed by Kozic &al to high spectral efficient modulation schemes. We study at first Chaos Coded Modulation based on the use of small dimensional modulo-MAP encoding process and we give a solution to study the distance spectrum of such coding schemes to accurately predict their performances. However, the obtained performances are quite poor. To improve them, we use then a high dimensional modulo-MAP mapping process similar to the low-density generator-matrix codes (LDGM) introduced by Kozic &al. The main difference with their work is that we use an encoding and decoding process on GF (2m) which enables to obtain better performances while preserving a quite simple decoding algorithm when we use the Extended Min-Sum (EMS) algorithm of Declercq &Fossorier.展开更多
This paper aims to explore a simpler and more user-friendly way of generating software based on model-driven development.Previous studies have attempted to generate code from domain models,hoping to reduce coding time...This paper aims to explore a simpler and more user-friendly way of generating software based on model-driven development.Previous studies have attempted to generate code from domain models,hoping to reduce coding time by increasing modeling time.However,as code tools become more advanced,it is challenging to improve efficiency because models are abstract while implementations are concrete.This paper proposes a novel approach that integrates ChatGPT as a plug-in into the whole R&D process and combines it with our code generation tool to enhance R&D efficiency.We have developed some demos to demonstrate the effectiveness of our approach.According to our evaluation,our approach can save more than 90%of the work in implementing the code generation tool,leaving only about 10%of the work for code review,code improvement,and unit testing.展开更多
This paper studies the nonsystematic Low-Density Parity-Check(LDPC)codes based onSymmetric Balanced Incomplete Block Design(SBIBD).First,it is concluded that the performancedegradation of nonsystematic linear block co...This paper studies the nonsystematic Low-Density Parity-Check(LDPC)codes based onSymmetric Balanced Incomplete Block Design(SBIBD).First,it is concluded that the performancedegradation of nonsystematic linear block codes is bounded by the average row weight of generalizedinverses of their generator matrices and code rate.Then a class of nonsystematic LDPC codes con-structed based on SBIBD is presented.Their characteristics include:both generator matrices andparity-check matrices are sparse and cyclic,which are simple to encode and decode;and almost arbi-trary rate codes can be easily constructed,so they are rate-compatible codes.Because there aresparse generalized inverses of generator matrices,the performance of the proposed codes is only0.15dB away from that of the traditional systematic LDPC codes.展开更多
文摘In this paper, the maximal length of maximal distance separable (MDS) codes is studied, and a new upper bound formula of the maximal length of MDS codes is obtained. Especially, the exact values of the maximal length of MDS codes in some parameters are given.
基金Supported by Foundation of Sichuan Tourism University(20SCTUTY01)Initial Scientific Research Fund of Doctors in Sichuan Tourism University。
文摘Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.
文摘In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe Chapote, these maximal codes are constructed over finite fields, and these codes are used for coding and decoding.
基金The National Natural Science Foundationof China (No.:60272048) Natural Science Foundationof JiangsuEducation Department(No.04kjb510057) China Scholarship Council
文摘Optical orthogonal code is the main signature code employed by optical CDMA system. Starting from modern mathematics theory, finite projective geometry and Galois theory, the essential connection between optical orthogonal code designing and finite geometry theory were discussed; find out the corresponding relationship between the parameter of OOC and that of finite geometry space. In this article, the systematic theory of OOC designing based on projective geometry is established in detail. The designing process and results of OOC on projective plane PG(2,q) and on m-dimension projective space are given respectively. Furthermore, the analytical theory for the corresponding relation between OOC with high cross-correlation and k-D manifold of projective space is set up. The OOC designing results given in this article have excellent performance, whose maximum cross-correlation is 1, and the cardinality reaches the Johnson upper bound, i.e. it realizes the optimization in both MUI and system capacity.
文摘Let E be an Archimedean Riesz algebra possessing a weak unit element e and a maximal disjoint system {e,: i∈I} in which e, is a projection element for each i. The principal band generated by eiis denoted by B(ei). The main result in this paper says that if there exists a completely regular Hausdorff space X such that E is Riesz algebra isomorphic to C(X) then for every i ∈ I there exists a completely regular Hausdorff space X, such that B(ei) is Riesz algebra isomorphic to C(Xi). Under an additional condition the inverse holds.
基金Foundation item： Supported by the Scientific Research Foundation of Hubei Provincial Education Depart- ment（B2013069） Supported by the National Science Foundation of Hubei Polytechnic University（12xjzl4A, 11yjz37B）
文摘The codes of formal power series rings R_∞=F[[r]]={sum from i=0 to ∞(a_lr^l|a_l∈F)}and finite chain rings R_i={a_0+a_1r+…+a_(i-1)r^(i-1)|a_i∈F}have close relationship in lifts and projection.In this paper,we study self-dual codes over R_∞by means of self-dual codes over Ri,and give some characterizations of self-dual codes over R_∞.
基金Supported by the National Natural Science Foundation of China(11861071).
文摘In this paper,we exhibit a free monoid containing all prefix codes in connection with the sets of i-th powers of primitive words for all i≥2.This extends two results given by Shyr and Tsai in 1998 at the same time.
基金financially supported in part by National Key R&D Program of China(No.2018YFB1801402)in part by Huawei Technologies Co.,Ltd.
文摘In this paper,we innovatively associate the mutual information with the frame error rate(FER)performance and propose novel quantized decoders for polar codes.Based on the optimal quantizer of binary-input discrete memoryless channels(BDMCs),the proposed decoders quantize the virtual subchannels of polar codes to maximize mutual information(MMI)between source bits and quantized symbols.The nested structure of polar codes ensures that the MMI quantization can be implemented stage by stage.Simulation results show that the proposed MMI decoders with 4 quantization bits outperform the existing nonuniform quantized decoders that minimize mean-squared error(MMSE)with 4 quantization bits,and yield even better performance than uniform MMI quantized decoders with 5 quantization bits.Furthermore,the proposed 5-bit quantized MMI decoders approach the floating-point decoders with negligible performance loss.
文摘Random Projection Code (RPC) is a mechanism that combines channel coding and modulation together and realizes rate adaptation in the receiving end. Random projection code’s mapping matrix has significant influences on decoding performance as well as hardware implementation complexity. To reduce hardware implementation complexity, we design a quasi-cyclic mapping matrix for RPC codes. Compared with other construction approaches, our design gets rid of data filter component, thus reducing chip area 7284.95 um2, and power consumption 331.46 uW in 0.13 um fabrication. Our simulation results show that our method does not cause any performance loss and even gets 0.2 dB to 0.5 dB gain at BER 10-4.
文摘The authors p oint out a problem in the article of Ref.(Xiong Hongyun,Rong Ximin.Maximal disjoint systems in Riesz space and representation.Acta Math Sinica ,1998,41(4):763-766.)and revise it.Let E be an Archimed ean Riesz space possessing a weak unit e and a maximal disjoint division{e i:i∈I} in which each e i is a proj ection element. Concerning the following statements:(1) There exists a completel y regular Hausdorff space X such that E is Riesz isomorphic to C(X);(2) For every i∈I there exi sts a completely regular Hausdorff space X i such that the band generated by e i is Riesz isomorphic to C(X i).It is shown that (1) implies (2),and find some conditions for the inverse bein g true are found.Furthermore,if each X i in (2) is a compact Ha usdorff space, a necessary and sufficient condition is established under which E can be represented as a C(X) for some compact Hausdorff space X.As corollaries, corresponding results for late rally complete Riesz spaces are obtained.
基金supported by MOST under Grant MOST 103-2633-E-242-002
文摘The most popular and representative classic waveform codes are referred to as orthogonal,bi-orthogonal,simplex,and etc,but the choice of waveform codes is essentially identical in error performance and cross correlation characteristic.Though bi-orthogonal coding requires half the bandwidth of the others,such coding scheme is attractive only when large bandwidth is available.In this paper,a novel finite projective plane（FPP） based waveform coding scheme is proposed,which is with similar error performance and cross correlation.Nevertheless,the bandwidth requirement will grow in a quadratic way,but not in an exponential way with the values of message bit numbers（k）.The proposed scheme takes obvious advantages over the bi-orthogonal scheme when k ≥ 6.
文摘In this paper, we propose to generalize the coding schemes first proposed by Kozic &al to high spectral efficient modulation schemes. We study at first Chaos Coded Modulation based on the use of small dimensional modulo-MAP encoding process and we give a solution to study the distance spectrum of such coding schemes to accurately predict their performances. However, the obtained performances are quite poor. To improve them, we use then a high dimensional modulo-MAP mapping process similar to the low-density generator-matrix codes (LDGM) introduced by Kozic &al. The main difference with their work is that we use an encoding and decoding process on GF (2m) which enables to obtain better performances while preserving a quite simple decoding algorithm when we use the Extended Min-Sum (EMS) algorithm of Declercq &Fossorier.
基金fully supported by the Natural Science Foundation of Hubei Province in China(Grant No.2021CFB482)Basic Research Science and Technology Project of Xiangyang(High-tech Domain 2022ABH007013)Hubei Superior and Distinctive Discipline Group of“New Energy Vehicle and Smart Transportation”。
文摘This paper aims to explore a simpler and more user-friendly way of generating software based on model-driven development.Previous studies have attempted to generate code from domain models,hoping to reduce coding time by increasing modeling time.However,as code tools become more advanced,it is challenging to improve efficiency because models are abstract while implementations are concrete.This paper proposes a novel approach that integrates ChatGPT as a plug-in into the whole R&D process and combines it with our code generation tool to enhance R&D efficiency.We have developed some demos to demonstrate the effectiveness of our approach.According to our evaluation,our approach can save more than 90%of the work in implementing the code generation tool,leaving only about 10%of the work for code review,code improvement,and unit testing.
基金the National Natural Science Foundation of China(No.60272009,No.60472045,and No.60496313).
文摘This paper studies the nonsystematic Low-Density Parity-Check(LDPC)codes based onSymmetric Balanced Incomplete Block Design(SBIBD).First,it is concluded that the performancedegradation of nonsystematic linear block codes is bounded by the average row weight of generalizedinverses of their generator matrices and code rate.Then a class of nonsystematic LDPC codes con-structed based on SBIBD is presented.Their characteristics include:both generator matrices andparity-check matrices are sparse and cyclic,which are simple to encode and decode;and almost arbi-trary rate codes can be easily constructed,so they are rate-compatible codes.Because there aresparse generalized inverses of generator matrices,the performance of the proposed codes is only0.15dB away from that of the traditional systematic LDPC codes.
