In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized...In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes.展开更多
A new method for constructing Quasi-Cyclic (QC) Low-Density Parity-Check (LDPC) codes based on Euclidean Geometry (EG) is presented. The proposed method results in a class of QC-LDPC codes with girth of at least 6 and...A new method for constructing Quasi-Cyclic (QC) Low-Density Parity-Check (LDPC) codes based on Euclidean Geometry (EG) is presented. The proposed method results in a class of QC-LDPC codes with girth of at least 6 and the designed codes perform very close to the Shannon limit with iterative decoding. Simulations show that the designed QC-LDPC codes have almost the same performance with the existing EG-LDPC codes.展开更多
In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power s...In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.展开更多
Permutation codes over finite chain rings are introduced; by using the character of the finite chain rings and the knowledge of representation of group, some conditions for existence or non-existence of self-dual perm...Permutation codes over finite chain rings are introduced; by using the character of the finite chain rings and the knowledge of representation of group, some conditions for existence or non-existence of self-dual permutation codes over finite chain rings are obtained. Specially, when the group is a direct product of a 2-group and a T-group, and the group action is transitive, the sufficient and necessary condition of the existence of permutation codes is given.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
The minimum squared Euclidean distance(MSED) of binary multi-h phase codes is presented. The signal segregation degree(SSD) has been put forward to determine MSED of multi-h phase codes. In order to maximize MSED, SSD...The minimum squared Euclidean distance(MSED) of binary multi-h phase codes is presented. The signal segregation degree(SSD) has been put forward to determine MSED of multi-h phase codes. In order to maximize MSED, SSD should be as large as possible. The necessary and sufficient conditions of maximizing SSD are derived. Finally, SSD and the exact formulae for MSED of binary 2-h phase codes are also presented.展开更多
The dual-containing (or self-orthogonal) formalism of Calderbank-Shor-Steane (CSS) codes provides a universal connection between a classical linear code and a Quantum Error-Correcting Code (QECC). We propose a novel c...The dual-containing (or self-orthogonal) formalism of Calderbank-Shor-Steane (CSS) codes provides a universal connection between a classical linear code and a Quantum Error-Correcting Code (QECC). We propose a novel class of quantum Low Density Parity Check (LDPC) codes constructed from cyclic classes of lines in Euclidean Geometry (EG). The corresponding constructed parity check matrix has quasi-cyclic structure that can be encoded flexibility, and satisfies the requirement of dual-containing quantum code. Taking the advantage of quasi-cyclic structure, we use a structured approach to construct Generalized Parity Check Matrix (GPCM). This new class of quantum codes has higher code rate, more sparse check matrix, and exactly one four-cycle in each pair of two rows. Ex-perimental results show that the proposed quantum codes, such as EG(2,q)II-QECC, EG(3,q)II-QECC, have better performance than that of other methods based on EG, over the depolarizing channel and decoded with iterative decoding based on the sum-product decoding algorithm.展开更多
A definition of a self-dual code on graph and a procedure based on factor graphs to judge a self-dual code were presented. Three contributions of this paper were described as follows. To begin with, transform T_ R→L ...A definition of a self-dual code on graph and a procedure based on factor graphs to judge a self-dual code were presented. Three contributions of this paper were described as follows. To begin with, transform T_ R→L were defined, which was the basis of self-dual codes defined on graphs and played a key role in the paper. The second were that a self-dual code could be defined on factor graph, which was much different from conventional algebraic method. The third was that a factor graph approach to judge a self-dual code was illustrated, which took advantage of duality properties of factor graphs and our proposed transform T_ R→L to offer a convenient and geometrically intuitive process to judge a self-dual code.展开更多
In this paper, we give an explicit construction for self-dual codes over F_p+vF_p(v^2= v) and determine all the self-dual codes over F_p+ vF_p by using self-dual codes over finite field F_p, where p is a prime.
In this paper,we first give the definition of the Euclidean sums of linear codes,and prove that the Euclidean sums of linear codes are Euclidean dual-containing.Then we construct two new classes of optimal asymmetric ...In this paper,we first give the definition of the Euclidean sums of linear codes,and prove that the Euclidean sums of linear codes are Euclidean dual-containing.Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes,and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields.Moreover,these optimal asymmetric quantum errorcorrecting codes constructed in this paper are different from the ones in the literature.展开更多
In this paper, we discuss a kind of Hermitian inner product - symplectic inner product, which is different from the original inner product - Euclidean inner product. According to the definition of symplectic inner pro...In this paper, we discuss a kind of Hermitian inner product - symplectic inner product, which is different from the original inner product - Euclidean inner product. According to the definition of symplectic inner product, the codes under the symplectic inner product have better properties than those under the general Hermitian inner product. Here we present the necessary and sufficient condition for judging whether a linear code C over Fp with a generator matrix in the standard form is a symplectic self-dual code. In addition, we give a method for constructing a new symplectic self-dual codes over Fp, which is simpler than others.展开更多
In Comparison with the traditional point-by-point line generation method,the method we present is based on segment code in Pan-Euclidean geometric space and is quite different in re- spect of running speed and theoret...In Comparison with the traditional point-by-point line generation method,the method we present is based on segment code in Pan-Euclidean geometric space and is quite different in re- spect of running speed and theoretical basis.This paper presents an approach of using segment code to draw straight lines and shows the characteristics of a digital line.It is a newly proposed al- gorithm applicable in CAD.展开更多
Two new design approaches for constructing Low-Density Parity-Check(LDPC) codes are proposed.One is used to design regular Quasi-Cyclic LDPC(QC-LDPC) codes with girth at least 8.The other is used to design irregular L...Two new design approaches for constructing Low-Density Parity-Check(LDPC) codes are proposed.One is used to design regular Quasi-Cyclic LDPC(QC-LDPC) codes with girth at least 8.The other is used to design irregular LDPC codes.Both of their parity-check matrices are composed of Circulant Permutation Matrices(CPMs).When iteratively decoded with the Sum-Product Algorithm(SPA),these proposed codes exhibit good performances over the AWGN channel.展开更多
It is a regular way of constructing quantum error-correcting codes via codes with self-orthogonal property, and whether a classical Bose-Chaudhuri-Hocquenghem (BCH) code is self-orthogonal can be determined by its des...It is a regular way of constructing quantum error-correcting codes via codes with self-orthogonal property, and whether a classical Bose-Chaudhuri-Hocquenghem (BCH) code is self-orthogonal can be determined by its designed distance. In this paper, we give the sufficient and necessary condition for arbitrary classical BCH codes with self-orthogonal property through algorithms. We also give a better upper bound of the designed distance of a classical narrow-sense BCH code which contains its Euclidean dual. Besides these, we also give one algorithm to compute the dimension of these codes. The complexity of all algorithms is analyzed. Then the results can be applied to construct a series of quantum BCH codes via the famous CSS constructions.展开更多
By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and suffi...By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.展开更多
Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to...Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.展开更多
Abstract:Sparse coding(SC)based visual tracking(l1-tracker)is gaining increasing attention,and many related algorithms are developed.In these algorithms,each candidate region is sparsely represented as a set of target...Abstract:Sparse coding(SC)based visual tracking(l1-tracker)is gaining increasing attention,and many related algorithms are developed.In these algorithms,each candidate region is sparsely represented as a set of target templates.However,the structure connecting these candidate regions is usually ignored.Lu proposed an NLSSC-tracker with non-local self-similarity sparse coding to address this issue,which has a high computational cost.In this study,we propose an Euclidean local-structure constraint based sparse coding tracker with a smoothed Euclidean local structure.With this tracker,the optimization procedure is transformed to a small-scale l1-optimization problem,significantly reducing the computational cost.Extensive experimental results on visual tracking demonstrate the eectiveness and efficiency of the proposed algorithm.展开更多
Let <i>f</i>(u) and <i>g</i>(v) be two polynomials of degree <i>k</i> and <i>l</i> respectively, not both linear which split into distinct linear factors over F<sub&g...Let <i>f</i>(u) and <i>g</i>(v) be two polynomials of degree <i>k</i> and <i>l</i> respectively, not both linear which split into distinct linear factors over F<sub>q</sub>. Let <img src="Edit_83041428-d8b0-4505-8c3c-5e29f2886159.png" width="160" height="15" alt="" /> be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring <i>R</i>. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from <img src="Edit_c75f119d-3176-4a71-a36a-354955044c09.png" width="50" height="15" alt="" /> which preserves duality. The Gray images of polyadic codes and their extensions over the ring <i>R</i> lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over F<i><sub>q</sub></i>. Some examples are also given to illustrate this.展开更多
Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) is a bandwidth ef- ficient transmission, where the bit error rate is reduced through the iterative information exchange between the inner demapper and...Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) is a bandwidth ef- ficient transmission, where the bit error rate is reduced through the iterative information exchange between the inner demapper and the outer decoder. The choice of the symbol mapping is the crucial design parameter. This paper indicates that the Harmonic Mean of the Minimum Squared Euclidean (HMMSE) distance is the best criterion for the mapping design. Based on the design criterion of the HMMSE distance, a new search algorithm to find the optimized labeling maps for BICM-ID system is proposed. Numerical results and performance comparison show that the new labeling search method has a low complexity and outperforms other labeling schemes using other design criterion in BICM-ID system, therefore it is an optimized labeling method.展开更多
文摘In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes.
基金Supported by the National Key Basic Research Program (973) Project (No. 2010CB328300)the 111 Project (No. B08038)
文摘A new method for constructing Quasi-Cyclic (QC) Low-Density Parity-Check (LDPC) codes based on Euclidean Geometry (EG) is presented. The proposed method results in a class of QC-LDPC codes with girth of at least 6 and the designed codes perform very close to the Shannon limit with iterative decoding. Simulations show that the designed QC-LDPC codes have almost the same performance with the existing EG-LDPC codes.
文摘In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.
基金Supported by the National Natural Science Foundation of China (60373087, 60473023, 90104005, 60673071)
文摘Permutation codes over finite chain rings are introduced; by using the character of the finite chain rings and the knowledge of representation of group, some conditions for existence or non-existence of self-dual permutation codes over finite chain rings are obtained. Specially, when the group is a direct product of a 2-group and a T-group, and the group action is transitive, the sufficient and necessary condition of the existence of permutation codes is given.
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_∞.
文摘The minimum squared Euclidean distance(MSED) of binary multi-h phase codes is presented. The signal segregation degree(SSD) has been put forward to determine MSED of multi-h phase codes. In order to maximize MSED, SSD should be as large as possible. The necessary and sufficient conditions of maximizing SSD are derived. Finally, SSD and the exact formulae for MSED of binary 2-h phase codes are also presented.
基金Supported by the National Natural Science Foundation ofChina (No. 61071145,41074090)the Specialized Research Fund for the Doctoral Program of Higher Education (200802880014)
文摘The dual-containing (or self-orthogonal) formalism of Calderbank-Shor-Steane (CSS) codes provides a universal connection between a classical linear code and a Quantum Error-Correcting Code (QECC). We propose a novel class of quantum Low Density Parity Check (LDPC) codes constructed from cyclic classes of lines in Euclidean Geometry (EG). The corresponding constructed parity check matrix has quasi-cyclic structure that can be encoded flexibility, and satisfies the requirement of dual-containing quantum code. Taking the advantage of quasi-cyclic structure, we use a structured approach to construct Generalized Parity Check Matrix (GPCM). This new class of quantum codes has higher code rate, more sparse check matrix, and exactly one four-cycle in each pair of two rows. Ex-perimental results show that the proposed quantum codes, such as EG(2,q)II-QECC, EG(3,q)II-QECC, have better performance than that of other methods based on EG, over the depolarizing channel and decoded with iterative decoding based on the sum-product decoding algorithm.
基金The National Natural Science Foundation of China (No60472018)
文摘A definition of a self-dual code on graph and a procedure based on factor graphs to judge a self-dual code were presented. Three contributions of this paper were described as follows. To begin with, transform T_ R→L were defined, which was the basis of self-dual codes defined on graphs and played a key role in the paper. The second were that a self-dual code could be defined on factor graph, which was much different from conventional algebraic method. The third was that a factor graph approach to judge a self-dual code was illustrated, which took advantage of duality properties of factor graphs and our proposed transform T_ R→L to offer a convenient and geometrically intuitive process to judge a self-dual code.
基金supported in part by the National Science Foundation of China under Grant 11571005in part by the Key Research Project of Higher Education of the Education Department of Henan Province under Grant 19A120010
文摘In this paper, we give an explicit construction for self-dual codes over F_p+vF_p(v^2= v) and determine all the self-dual codes over F_p+ vF_p by using self-dual codes over finite field F_p, where p is a prime.
基金Supported by the Scientific Research Foundation of Hubei Provincial Education Department of China(Q20174503)the National Science Foundation of Hubei Polytechnic University of China(12xjz14A and 17xjz03A)。
文摘In this paper,we first give the definition of the Euclidean sums of linear codes,and prove that the Euclidean sums of linear codes are Euclidean dual-containing.Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes,and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields.Moreover,these optimal asymmetric quantum errorcorrecting codes constructed in this paper are different from the ones in the literature.
文摘In this paper, we discuss a kind of Hermitian inner product - symplectic inner product, which is different from the original inner product - Euclidean inner product. According to the definition of symplectic inner product, the codes under the symplectic inner product have better properties than those under the general Hermitian inner product. Here we present the necessary and sufficient condition for judging whether a linear code C over Fp with a generator matrix in the standard form is a symplectic self-dual code. In addition, we give a method for constructing a new symplectic self-dual codes over Fp, which is simpler than others.
文摘In Comparison with the traditional point-by-point line generation method,the method we present is based on segment code in Pan-Euclidean geometric space and is quite different in re- spect of running speed and theoretical basis.This paper presents an approach of using segment code to draw straight lines and shows the characteristics of a digital line.It is a newly proposed al- gorithm applicable in CAD.
基金Supported by the National Natural Science Foundation of China(Nos.61271199,61172022)
文摘Two new design approaches for constructing Low-Density Parity-Check(LDPC) codes are proposed.One is used to design regular Quasi-Cyclic LDPC(QC-LDPC) codes with girth at least 8.The other is used to design irregular LDPC codes.Both of their parity-check matrices are composed of Circulant Permutation Matrices(CPMs).When iteratively decoded with the Sum-Product Algorithm(SPA),these proposed codes exhibit good performances over the AWGN channel.
基金Supported by the National Natural Science Foundation of China (No.60403004)the Outstanding Youth Foundation of China (No.0612000500)
文摘It is a regular way of constructing quantum error-correcting codes via codes with self-orthogonal property, and whether a classical Bose-Chaudhuri-Hocquenghem (BCH) code is self-orthogonal can be determined by its designed distance. In this paper, we give the sufficient and necessary condition for arbitrary classical BCH codes with self-orthogonal property through algorithms. We also give a better upper bound of the designed distance of a classical narrow-sense BCH code which contains its Euclidean dual. Besides these, we also give one algorithm to compute the dimension of these codes. The complexity of all algorithms is analyzed. Then the results can be applied to construct a series of quantum BCH codes via the famous CSS constructions.
基金Supported by the National Natural Science Foundation of China(No.61370089)
文摘By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.
文摘Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.
基金National Natural Foundation of China under Grant(61572085,61502058)
文摘Abstract:Sparse coding(SC)based visual tracking(l1-tracker)is gaining increasing attention,and many related algorithms are developed.In these algorithms,each candidate region is sparsely represented as a set of target templates.However,the structure connecting these candidate regions is usually ignored.Lu proposed an NLSSC-tracker with non-local self-similarity sparse coding to address this issue,which has a high computational cost.In this study,we propose an Euclidean local-structure constraint based sparse coding tracker with a smoothed Euclidean local structure.With this tracker,the optimization procedure is transformed to a small-scale l1-optimization problem,significantly reducing the computational cost.Extensive experimental results on visual tracking demonstrate the eectiveness and efficiency of the proposed algorithm.
文摘Let <i>f</i>(u) and <i>g</i>(v) be two polynomials of degree <i>k</i> and <i>l</i> respectively, not both linear which split into distinct linear factors over F<sub>q</sub>. Let <img src="Edit_83041428-d8b0-4505-8c3c-5e29f2886159.png" width="160" height="15" alt="" /> be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring <i>R</i>. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from <img src="Edit_c75f119d-3176-4a71-a36a-354955044c09.png" width="50" height="15" alt="" /> which preserves duality. The Gray images of polyadic codes and their extensions over the ring <i>R</i> lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over F<i><sub>q</sub></i>. Some examples are also given to illustrate this.
基金Supported by the National Natural Science Foundation of China (No.60472104).
文摘Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) is a bandwidth ef- ficient transmission, where the bit error rate is reduced through the iterative information exchange between the inner demapper and the outer decoder. The choice of the symbol mapping is the crucial design parameter. This paper indicates that the Harmonic Mean of the Minimum Squared Euclidean (HMMSE) distance is the best criterion for the mapping design. Based on the design criterion of the HMMSE distance, a new search algorithm to find the optimized labeling maps for BICM-ID system is proposed. Numerical results and performance comparison show that the new labeling search method has a low complexity and outperforms other labeling schemes using other design criterion in BICM-ID system, therefore it is an optimized labeling method.
