The Experts below are selected from a list of 28518 Experts worldwide ranked by ideXlab platform
Khaled Abdelghaffar - One of the best experts on this subject based on the ideXlab platform.
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correcting combinations of errors and erasures with Euclidean Geometry ldpc codes
International Symposium on Information Theory, 2013Co-Authors: Qiuju Diao, Shu Lin, Ying Yu Tai, Khaled AbdelghaffarAbstract:It is shown that Euclidean Geometry LDPC codes in conjunction with their shortened codes obtained by puncturing their parity-check matrices are effective in correcting combinations of errors and erasures with a two-phase decoding scheme. This is due to the large row redundancies of the parity-check matrices of these codes which are given by the incidence matrices of Euclidean geometries.
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a transform approach for constructing quasi cyclic Euclidean Geometry ldpc codes
Information Theory and Applications, 2012Co-Authors: Qiuju Diao, Wei Zhou, Shu Lin, Khaled AbdelghaffarAbstract:A method for constructing quasi-cyclic Euclidean Geometry (QC-EG) LDPC codes in the Fourier transform domain is presented. Given a Euclidean Geometry over a finite field of characteristic 2, base matrices in the Fourier transform domain are first constructed. Then the inverse Fourier transforms of these base matrices, combined with row and column permutations, result in low-density arrays of circulant permutation matrices and/or zero matrices. The null spaces of these low-density arrays give a family of QC-EG-LDPC codes. Codes in a special subclass have large minimum distances and their Tanner graphs contain no harmful trapping sets with sizes smaller than their minimum distances.
Richard V Kadison - One of the best experts on this subject based on the ideXlab platform.
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the pythagorean theorem ii the infinite discrete case
Proceedings of the National Academy of Sciences of the United States of America, 2002Co-Authors: Richard V KadisonAbstract:The study of the Pythagorean Theorem and variants of it as the basic result of noncommutative, metric, Euclidean Geometry is continued. The emphasis in the present article is the case of infinite discrete dimensionality.
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the pythagorean theorem i the finite case
Proceedings of the National Academy of Sciences of the United States of America, 2002Co-Authors: Richard V KadisonAbstract:The Pythagorean Theorem and variants of it are studied. The variations evolve to a formulation in terms of noncommutative, conditional expectations on von Neumann algebras that displays the theorem as the basic result of noncommutative, metric, Euclidean Geometry. The emphasis in the present article is finite dimensionality, both “discrete” and “continuous.”
Charles Gunn - One of the best experts on this subject based on the ideXlab platform.
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geometric algebras for Euclidean Geometry
Advances in Applied Clifford Algebras, 2017Co-Authors: Charles GunnAbstract:The discussion of how to apply geometric algebra to Euclidean $${n}$$ -space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from nineteenth century mathematics. We then introduce the dual projectivized Clifford algebra $${\mathbf{P}(\mathbb{R}_{n,0,1}^{*})}$$ (Euclidean PGA) as the most promising homogeneous (1-up) candidate for Euclidean Geometry. We compare Euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that Euclidean PGA is the smallest structure-preserving Euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include Euclidean sphere primitives. We conclude that Euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.
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Geometric Algebras for Euclidean Geometry
Advances in Applied Clifford Algebras, 2017Co-Authors: Charles GunnAbstract:The discussion of how to apply geometric algebra to Euclidean $n$-space has been clouded by a number of conceptual misunderstandings which we first identify and resolve, based on a thorough review of crucial but largely forgotten themes from $19^{th}$ century mathematics. We then introduce the dual projectivized Clifford algebra $\mathbf{P}(\mathbb{R}^*_{n,0,1})$ (Euclidean PGA) as the most promising homogeneous (1-up) candidate for Euclidean Geometry. We compare Euclidean PGA and the popular 2-up model CGA (conformal geometric algebra), restricting attention to flat geometric primitives, and show that on this domain they exhibit the same formal feature set. We thereby establish that Euclidean PGA is the smallest structure-preserving Euclidean GA. We compare the two algebras in more detail, with respect to a number of practical criteria, including implementation of kinematics and rigid body mechanics. We then extend the comparison to include Euclidean sphere primitives. We conclude that Euclidean PGA provides a natural transition, both scientifically and pedagogically, between vector space models and the more complex and powerful CGA.
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on the homogeneous model of Euclidean Geometry
arXiv: Metric Geometry, 2011Co-Authors: Charles GunnAbstract:We attach the degenerate signature (n,0,1) to the projectivized dual Grassmann algebra over R(n+1). We explore the use of the resulting Clifford algebra as a model for Euclidean Geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between this Grassmann algebra and its dual, that yields non-metric meet and join operators. We review the Cayley-Klein construction of the projective (homogeneous) model for Euclidean Geometry leading to the choice of the signature (n,0,1). We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between simple k- and m-vectors. We establish that versor (sandwich) operators provide all Euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct Euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of such elements. We conclude with an elementary account of Euclidean rigid body motion within this framework.
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on the homogeneous model of Euclidean Geometry
Guide to Geometric Algebra in Practice, 2011Co-Authors: Charles GunnAbstract:We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for Euclidean Geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between k-blades and m-blades. We identify sandwich operators in the algebra that provide all Euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct Euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of group elements. We conclude with an elementary account of Euclidean kinematics and rigid body motion within this framework.
Jinhong Yuan - One of the best experts on this subject based on the ideXlab platform.
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Euclidean Geometry based spatially coupled ldpc codes for storage
IEEE Journal on Selected Areas in Communications, 2016Co-Authors: Yixuan Xie, Lei Yang, Peng Kang, Jinhong YuanAbstract:In this paper, we construct binary spatially coupled (SC) low-density parity-check (LDPC) codes based on Euclidean Geometry (EG) LDPC codes for storage applications, where high error correction capability, extremely low uncorrectable bit error rate (UBER), and low decoding complexity are required. We propose a systematic way to construct the families of SC LDPC codes from $(m,2^{s})$ EG LDPC codes, which are termed EG-SC LDPC codes. In the construction method, we propose a 2-D edge-spreading process to construct the base matrix of EG-SC LDPC codes, which consists of matrix unwrapping and periodically time-varying of a protograph. A lower bound on the rank of the parity-check matrix of an EG-SC LDPC code is derived. We evaluate the error rate performance of the constructed EG-SC LDPC codes by using a weighted bit-flipping decoding algorithm for its low decoding complexity. Numerical results show that the UBER performance of the constructed EG-SC LDPC codes is superior to that of their EG LDPC code counterparts, and show no error floor compared with the constructed protograph SC LDPC codes and regular LDPC codes.
Qiuju Diao - One of the best experts on this subject based on the ideXlab platform.
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correcting combinations of errors and erasures with Euclidean Geometry ldpc codes
International Symposium on Information Theory, 2013Co-Authors: Qiuju Diao, Shu Lin, Ying Yu Tai, Khaled AbdelghaffarAbstract:It is shown that Euclidean Geometry LDPC codes in conjunction with their shortened codes obtained by puncturing their parity-check matrices are effective in correcting combinations of errors and erasures with a two-phase decoding scheme. This is due to the large row redundancies of the parity-check matrices of these codes which are given by the incidence matrices of Euclidean geometries.
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a transform approach for constructing quasi cyclic Euclidean Geometry ldpc codes
Information Theory and Applications, 2012Co-Authors: Qiuju Diao, Wei Zhou, Shu Lin, Khaled AbdelghaffarAbstract:A method for constructing quasi-cyclic Euclidean Geometry (QC-EG) LDPC codes in the Fourier transform domain is presented. Given a Euclidean Geometry over a finite field of characteristic 2, base matrices in the Fourier transform domain are first constructed. Then the inverse Fourier transforms of these base matrices, combined with row and column permutations, result in low-density arrays of circulant permutation matrices and/or zero matrices. The null spaces of these low-density arrays give a family of QC-EG-LDPC codes. Codes in a special subclass have large minimum distances and their Tanner graphs contain no harmful trapping sets with sizes smaller than their minimum distances.
