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 wireless multicast, network coding has recently attracted attentions as a substantial improvement to packet retransmission schemes. However, the problem of finding the optimal network code which minimizes the retra...In wireless multicast, network coding has recently attracted attentions as a substantial improvement to packet retransmission schemes. However, the problem of finding the optimal network code which minimizes the retransmissions is hard to solve or approximate. This paper presents two schemes to reduce the number of retransmissions for reliable multicast efficiently. One is retransmission using network coding based on improved Vandermonde matrix （VRNC）, the other is retransmission using network coding based on adaptive improved Vandermonde matrix （AVRNC）. Using VRNC scheme the sender selects the packets all receivers have lost and encodes them with improved Vandermonde matrix; when receivers receive enough encoded retransmission packets, all the lost packets can be recovered. With AVRNC scheme, the sender can obtain the recovery information from all the receivers after sending out per retransmission packet, and then the improved Vandermonde matrix can be updated, thus reducing the complexity of encoding and decoding. Our proposed schemes can achieve the theoretical lower bound assuming retransmission packets lossless, and approach the theoretical lower bound considering retransmission packets loss. Simulation results show that the proposed algorithms can efficiently reduce the number ofretransmissions, thus improving transmission efficiency.展开更多
We extend Vandermonde matrices to generalized Vandermonde tensors. We call an ruth order n-dimensional real tensor A = （Ai1i2…im） a type-1 generalized Vandermonde （GV） tensor, or GV1 tensor, if there exists a vec...We extend Vandermonde matrices to generalized Vandermonde tensors. We call an ruth order n-dimensional real tensor A = （Ai1i2…im） a type-1 generalized Vandermonde （GV） tensor, or GV1 tensor, if there exists a vector v = （v1, v2,.. , Vn）T such that Aili2...im = vi2＋i3＋＋im-m＋l and call A , a type-2 （ruth order n dimensional） GV tensor, or GV2 tensor, if there exists an （m - 1）th order tensor B= （Bi1i2…im-1） such that Ai1i2…im= Bim-1i1i2…im In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.展开更多
This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). ...This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). This node configuration can be considered to be a kind of extension of the Cross Type Node Configuration , in R 2 to high dimensional spaces. And the Mixed Type Node Configuration in R s(s>2) is also discussed in this paper in an example.展开更多
In this paper, we consider the Straight Line Type Node Configuration C （SLTNCC） in multivariate polynomial interpolation as the result of different kinds of transformations of lines （such as parallel translations, ...In this paper, we consider the Straight Line Type Node Configuration C （SLTNCC） in multivariate polynomial interpolation as the result of different kinds of transformations of lines （such as parallel translations, rotations）. Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.展开更多
We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z...We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.展开更多
In order to solve high encoding complexities of irregular low-density parity-check （LDPC） codes, a deterministic construction of irregular LDPC codes with low encoding complexities is proposed. The encoding algorith...In order to solve high encoding complexities of irregular low-density parity-check （LDPC） codes, a deterministic construction of irregular LDPC codes with low encoding complexities is proposed. The encoding algorithms are designed, whose complexities are linear equations of code length. The construction and encoding algorithms are derived from the effectively encoding characteristics of repeat-accumulate （RA） codes and masking technique. First, the new construction modifies parity-check matrices of RA codes to eliminate error floors of RA codes. Second, the new constructed parity-check matrices are based on Vandermonde matrices; this deterministic algebraic structure is easy for hardware implementation. Theoretic analysis and experimental results show that, at a bit-error rate of 10 × 10^-4, the new codes with lower encoding complexities outperform Mackay＇s random LDPC codes by 0.4-0.6 dB over an additive white Gauss noise （AWGN） channel.展开更多
EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn i...EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.展开更多
Network coding is able to address output conflicts when fanout splitting is allowed for multicast switching.Hence,it successfully achieves a larger rate region than non-coding approaches in crossbar switches.However,n...Network coding is able to address output conflicts when fanout splitting is allowed for multicast switching.Hence,it successfully achieves a larger rate region than non-coding approaches in crossbar switches.However,network coding requires large coding buffers and a high computational cost on encoding and decoding.In this paper,we propose a novel Online Network Coding framework called Online NC for multicast switches,which is adaptive to constrained buffers.Moreover,it enjoys a much lower decoding complexity by a Vandermonde matrix based approach,as compared to conven-tional randomized network coding Our approach realizes online coding with one coding algo-rithm that synchronizes buffering and coding.Therefore,we significantly reduce requirements on buffer space,while also sustaining high throughputs.We confirm the superior advantages of our contributions using empirical studies.展开更多
<div style="text-align:justify;"> <span style="font-family:Verdana;">In this paper using elementary Galois Theory, we give a detailed explanation of the calculation of the radical expre...<div style="text-align:justify;"> <span style="font-family:Verdana;">In this paper using elementary Galois Theory, we give a detailed explanation of the calculation of the radical expression for <img alt="" src="Edit_fd040e3d-ec1e-440c-a4c5-89b6c55a4a78.png" />which was first discussed by Vandermonde decades before Galois and we point out and correct a minor correction in his work which was also observed by Lagrange.</span> </div>展开更多
Suppose that x is a complex number and i is a non negative integer. Define N - i(x)=|x| i if i is even and N - i(x)=x|x| i-1 if i is odd. Let V - n(x 1, ...,x n) denote the ...Suppose that x is a complex number and i is a non negative integer. Define N - i(x)=|x| i if i is even and N - i(x)=x|x| i-1 if i is odd. Let V - n(x 1, ...,x n) denote the n× n matrix whose (i,j) th entry is N - i-1 (x j) . This paper presents a computation formula for det V - n(x 1, ...,x n) , which can be considered as a generalized that of Vandermonde determinant, and some its important theoretical applications.展开更多
Based on the double determinant theory the problem about the determinant of Vandermonde's type over quaternion field is studied, and a necessary and sufficient condition that this double determinant is not equal t...Based on the double determinant theory the problem about the determinant of Vandermonde's type over quaternion field is studied, and a necessary and sufficient condition that this double determinant is not equal to zero is got.展开更多
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.
基金supported by the Canada-China Scientific and Technological Cooperation (2010DFA11320)the Fundamental Research Funds for the Central Universities (G470209, 2009RC0308)+1 种基金the National Natural Science Foundation of China (60802033, 60873190)the Important National Science and Technology Specific Projects (2010ZX03007-003-04, 2010ZX03005-001-03)
文摘In wireless multicast, network coding has recently attracted attentions as a substantial improvement to packet retransmission schemes. However, the problem of finding the optimal network code which minimizes the retransmissions is hard to solve or approximate. This paper presents two schemes to reduce the number of retransmissions for reliable multicast efficiently. One is retransmission using network coding based on improved Vandermonde matrix （VRNC）, the other is retransmission using network coding based on adaptive improved Vandermonde matrix （AVRNC）. Using VRNC scheme the sender selects the packets all receivers have lost and encodes them with improved Vandermonde matrix; when receivers receive enough encoded retransmission packets, all the lost packets can be recovered. With AVRNC scheme, the sender can obtain the recovery information from all the receivers after sending out per retransmission packet, and then the improved Vandermonde matrix can be updated, thus reducing the complexity of encoding and decoding. Our proposed schemes can achieve the theoretical lower bound assuming retransmission packets lossless, and approach the theoretical lower bound considering retransmission packets loss. Simulation results show that the proposed algorithms can efficiently reduce the number ofretransmissions, thus improving transmission efficiency.
文摘We extend Vandermonde matrices to generalized Vandermonde tensors. We call an ruth order n-dimensional real tensor A = （Ai1i2…im） a type-1 generalized Vandermonde （GV） tensor, or GV1 tensor, if there exists a vector v = （v1, v2,.. , Vn）T such that Aili2...im = vi2＋i3＋＋im-m＋l and call A , a type-2 （ruth order n dimensional） GV tensor, or GV2 tensor, if there exists an （m - 1）th order tensor B= （Bi1i2…im-1） such that Ai1i2…im= Bim-1i1i2…im In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.
文摘This paper introduces the definition of the Orthogonal Type Node Configuration and discusses the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation problems in high dimensional space R s(s>2). This node configuration can be considered to be a kind of extension of the Cross Type Node Configuration , in R 2 to high dimensional spaces. And the Mixed Type Node Configuration in R s(s>2) is also discussed in this paper in an example.
文摘In this paper, we consider the Straight Line Type Node Configuration C （SLTNCC） in multivariate polynomial interpolation as the result of different kinds of transformations of lines （such as parallel translations, rotations）. Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.
文摘We generalize the Eulerian numbers ?to sets of numbers Eμ(k,l), (μ=0,1,2,···) where the Eulerian numbers appear as the special case μ=1. This can be used for the evaluation of generalizations Eμ(k,Z) of the Geometric series G0(k;Z)=G1(0;Z) by splitting an essential part (1-Z)-(μK+1) where the numbers Eμ(k,l) are then the coefficients of the remainder polynomial. This can be extended for non-integer parameter k to the approximative evaluation of generalized Geometric series. The recurrence relations and for the Generalized Eulerian numbers E1(k,l) are derived. The Eulerian numbers are related to the Stirling numbers of second kind S(k,l) and we give proofs for the explicit relations of Eulerian to Stirling numbers of second kind in both directions. We discuss some ordering relations for differentiation and multiplication operators which play a role in our derivations and collect this in Appendices.
基金Supported by the National Natural Science Foundation of China(60496315, 60572050)
文摘In order to solve high encoding complexities of irregular low-density parity-check （LDPC） codes, a deterministic construction of irregular LDPC codes with low encoding complexities is proposed. The encoding algorithms are designed, whose complexities are linear equations of code length. The construction and encoding algorithms are derived from the effectively encoding characteristics of repeat-accumulate （RA） codes and masking technique. First, the new construction modifies parity-check matrices of RA codes to eliminate error floors of RA codes. Second, the new constructed parity-check matrices are based on Vandermonde matrices; this deterministic algebraic structure is easy for hardware implementation. Theoretic analysis and experimental results show that, at a bit-error rate of 10 × 10^-4, the new codes with lower encoding complexities outperform Mackay＇s random LDPC codes by 0.4-0.6 dB over an additive white Gauss noise （AWGN） channel.
基金the NSF of Gansu Province of China (3ZS041-A25-007)
文摘EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.
基金Supported by the National 863 Projects of China(2009AA01Z205)the Fund of National Laboratory(P080010)+2 种基金the Natural Science Foundation of China(60872010,60972016)the Program for New Century Excellent Talents in University (NCET070339)the Funds for Distinguished Young Scientists of Hubei,China(2009 CDA150)
文摘Network coding is able to address output conflicts when fanout splitting is allowed for multicast switching.Hence,it successfully achieves a larger rate region than non-coding approaches in crossbar switches.However,network coding requires large coding buffers and a high computational cost on encoding and decoding.In this paper,we propose a novel Online Network Coding framework called Online NC for multicast switches,which is adaptive to constrained buffers.Moreover,it enjoys a much lower decoding complexity by a Vandermonde matrix based approach,as compared to conven-tional randomized network coding Our approach realizes online coding with one coding algo-rithm that synchronizes buffering and coding.Therefore,we significantly reduce requirements on buffer space,while also sustaining high throughputs.We confirm the superior advantages of our contributions using empirical studies.
文摘<div style="text-align:justify;"> <span style="font-family:Verdana;">In this paper using elementary Galois Theory, we give a detailed explanation of the calculation of the radical expression for <img alt="" src="Edit_fd040e3d-ec1e-440c-a4c5-89b6c55a4a78.png" />which was first discussed by Vandermonde decades before Galois and we point out and correct a minor correction in his work which was also observed by Lagrange.</span> </div>
文摘Suppose that x is a complex number and i is a non negative integer. Define N - i(x)=|x| i if i is even and N - i(x)=x|x| i-1 if i is odd. Let V - n(x 1, ...,x n) denote the n× n matrix whose (i,j) th entry is N - i-1 (x j) . This paper presents a computation formula for det V - n(x 1, ...,x n) , which can be considered as a generalized that of Vandermonde determinant, and some its important theoretical applications.
文摘Based on the double determinant theory the problem about the determinant of Vandermonde's type over quaternion field is studied, and a necessary and sufficient condition that this double determinant is not equal to zero is got.
