Computational Consierations In Network Coding On Plucker Embedded Subspace Codes

This book, written by experts from universities and major research laboratories, addresses the hot topic of network coding, a powerful scheme for information transmission in networks that yields near-optimal throughput. It introduces readers to this striking new approach to network coding, in which the network is not simply viewed as a mechanism for delivering packets, but rather an algebraic structure named the subspace, which these packets span. This leads to a new kind of coding theory, employing what are called subspace codes. The book presents selected, highly relevant advanced research output on: Subspace Codes and Rank Metric Codes; Finite Geometries and Subspace Designs; Application of Network Coding; Codes for Distributed Storage Systems. The outcomes reflect research conducted within the framework of the European COST Action IC1104: Random Network Coding and Designs over GF(q). Taken together, they offer communications engineers, R&D engineers, researchers and graduate students in Mathematics, Computer Science, and Electrical Engineering a comprehensive reference guide to the construction of optimal network codes, as well as efficient encoding and decoding schemes for a given network code.

This book, written by experts from universities and major research laboratories, addresses the hot topic of network coding, a powerful scheme for information transmission in networks that yields near-optimal throughput. It introduces readers to this striking new approach to network coding, in which the network is not simply viewed as a mechanism for delivering packets, but rather an algebraic structure named the subspace, which these packets span. This leads to a new kind of coding theory, employing what are called subspace codes. The book presents selected, highly relevant advanced research output on: Subspace Codes and Rank Metric Codes; Finite Geometries and Subspace Designs; Application of Network Coding; Codes for Distributed Storage Systems. The outcomes reflect research conducted within the framework of the European COST Action IC1104: Random Network Coding and Designs over GF(q). Taken together, they offer communications engineers, R & D engineers, researchers and graduate students in Mathematics, Computer Science, and Electrical Engineering a comprehensive reference guide to the construction of optimal network codes, as well as efficient encoding and decoding schemes for a given network code.

Subspace codes and rank-metric codes can be used to correct errors and erasures in networks with linear network coding. Both types of codes have been extensively studied in the past five years. We develop in this document list-decoding algorithms for subspace codes and rank-metric codes, thereby providing a better tradeoff between rate and error-correction capability than existing constructions. Randomized linear network coding, considered as the most practical approach to network coding, is a powerful tool for disseminating information in networks. Yet it is highly susceptible to transmission errors caused by noise or intentional jamming. Subspace codes were introduced by Koetter and Kschischang to correct errors and erasures in networks with a randomized protocol where the topology is unknown (the non-coherent case). The codewords of a subspace code are vector subspaces of a fixed ambient space; thus the codes are collections of such subspaces. We first develop a family of subspace codes, based upon the Koetter-Kschichang construction, which are efficiently list decodable. We show that, for a certain range of code rates, our list-decoding algorithm provides a better tradeoff between rate and decoding radius than the Koetter-Kschischang codes. We further improve these results by introducing multiple roots in the interpolation step of our list-decoding algorithm. To this end, we establish the notion of derivative and multiplicity in the ring of linearized polynomials. In order to achieve a better decoding radius, we take advantage of enforcing multiple roots for the interpolation polynomial. We are also able to list decode for a wider range of rates. Furthermore, we propose an alternative approach which leads to a linear-algebraic list-decoding algorithm. Rank-metric codes are suitable for error correction in the case where the network topology and the underlying network code are known (the coherent case). Gabidulin codes are a well-known class of algebraic rank-metric codes that meet the Singleton bound on the minimum rank-distance of a code. In this dissertation, we introduce a folded version of Gabidulin codes along with a list-decoding algorithm for such codes. Our list-decoding algorithm makes it possible to achieve the information theoretic bound on the decoding radius of a rank-metric code.

This book presents algorithmic tools for algebraic geometry, with experimental applications. It also introduces Macaulay 2, a computer algebra system supporting research in algebraic geometry, commutative algebra, and their applications. The algorithmic tools presented here are designed to serve readers wishing to bring such tools to bear on their own problems. The first part of the book covers Macaulay 2 using concrete applications; the second emphasizes details of the mathematics.

Projective geometry is not only a jewel of mathematics, but has also many applications in modern information and communication science. This book presents the foundations of classical projective and affine geometry as well as its important applications in coding theory and cryptography. It also could serve as a first acquaintance with diagram geometry. Written in clear and contemporary language with an entertaining style and around 200 exercises, examples and hints, this book is ideally suited to be used as a textbook for study in the classroom or on its own.

3264, the mathematical solution to a question concerning geometric figures.

A collection of surveys and research papers on mathematical software and algorithms. The common thread is that the field of mathematical applications lies on the border between algebra and geometry. Topics include polyhedral geometry, elimination theory, algebraic surfaces, Gröbner bases, triangulations of point sets and the mutual relationship. This diversity is accompanied by the abundance of available software systems which often handle only special mathematical aspects. This is why the volume also focuses on solutions to the integration of mathematical software systems. This includes low-level and XML based high-level communication channels as well as general frameworks for modular systems.

The leading mind behind the mathematics of string theory discusses how geometry explains the universe we see. Illustrations.

Did you know that any straight-line drawing on paper can be folded so that the complete drawing can be cut out with one straight scissors cut? That there is a planar linkage that can trace out any algebraic curve, or even 'sign your name'? Or that a 'Latin cross' unfolding of a cube can be refolded to 23 different convex polyhedra? Over the past decade, there has been a surge of interest in such problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this treatment gives hundreds of results and over 60 unsolved 'open problems' to inspire further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Aimed at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from school students to researchers.