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Author: Reza Dastbasteh Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
Quantum error-correcting codes (quantum codes) are applied to protect quantum information from errors caused by noise (decoherence) on the quantum channel in a way that is similar to that of classical error-correcting codes. The stabilizer construction is currently the most successful and widely used technique for constructing binary quantum codes. We explore new frontiers beyond the stabilizer construction. Our approach enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Our construction is particularly well-suited to certain families of classical codes, including duadic codes and additive twisted codes. For duadic codes, we provide various modifications of our construction and develop new computational strategies to bound the minimum distance. This enabled us to extend the tables of good duadic codes to much larger block lengths. The primary focus of this thesis is on additive twisted codes, which are highly structured but also technically much more difficult than the more common families of codes. They are widely referenced but have received relatively little development in previous studies. We discover new connections between twisted codes and linear cyclic codes and provide novel lower and upper bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann-Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. Another important contribution is the development of new criteria for code equivalence within the families of linear cyclic, constacyclic, and twisted codes. We introduce novel sufficient conditions for code equivalence and classify all equivalent codes of certain lengths. We prove a recent conjecture on a necessary condition for the formula describing affine equivalence. For twisted codes, we use algebraic methods, such as group actions, to determine many codes with the same parameters. These results have practical implications, as they are useful for pruning the search for new good codes, and they enabled us to discover many new record-breaking linear and binary quantum codes.
Author: Reza Dastbasteh Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
Quantum error-correcting codes (quantum codes) are applied to protect quantum information from errors caused by noise (decoherence) on the quantum channel in a way that is similar to that of classical error-correcting codes. The stabilizer construction is currently the most successful and widely used technique for constructing binary quantum codes. We explore new frontiers beyond the stabilizer construction. Our approach enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Our construction is particularly well-suited to certain families of classical codes, including duadic codes and additive twisted codes. For duadic codes, we provide various modifications of our construction and develop new computational strategies to bound the minimum distance. This enabled us to extend the tables of good duadic codes to much larger block lengths. The primary focus of this thesis is on additive twisted codes, which are highly structured but also technically much more difficult than the more common families of codes. They are widely referenced but have received relatively little development in previous studies. We discover new connections between twisted codes and linear cyclic codes and provide novel lower and upper bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann-Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. Another important contribution is the development of new criteria for code equivalence within the families of linear cyclic, constacyclic, and twisted codes. We introduce novel sufficient conditions for code equivalence and classify all equivalent codes of certain lengths. We prove a recent conjecture on a necessary condition for the formula describing affine equivalence. For twisted codes, we use algebraic methods, such as group actions, to determine many codes with the same parameters. These results have practical implications, as they are useful for pruning the search for new good codes, and they enabled us to discover many new record-breaking linear and binary quantum codes.
Author: Santosh Kumar Publisher: ISBN: Category : Languages : en Pages :
Book Description
The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between self-orthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.
Author: Avanti Ulhas Ketkar Publisher: ISBN: Category : Languages : en Pages :
Book Description
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
Author: Martin Tomlinson Publisher: Springer ISBN: 3319511033 Category : Technology & Engineering Languages : en Pages : 527
Book Description
This book discusses both the theory and practical applications of self-correcting data, commonly known as error-correcting codes. The applications included demonstrate the importance of these codes in a wide range of everyday technologies, from smartphones to secure communications and transactions. Written in a readily understandable style, the book presents the authors’ twenty-five years of research organized into five parts: Part I is concerned with the theoretical performance attainable by using error correcting codes to achieve communications efficiency in digital communications systems. Part II explores the construction of error-correcting codes and explains the different families of codes and how they are designed. Techniques are described for producing the very best codes. Part III addresses the analysis of low-density parity-check (LDPC) codes, primarily to calculate their stopping sets and low-weight codeword spectrum which determines the performance of th ese codes. Part IV deals with decoders designed to realize optimum performance. Part V describes applications which include combined error correction and detection, public key cryptography using Goppa codes, correcting errors in passwords and watermarking. This book is a valuable resource for anyone interested in error-correcting codes and their applications, ranging from non-experts to professionals at the forefront of research in their field. This book is open access under a CC BY 4.0 license.
Author: Gabriele Nebe Publisher: Springer Science & Business Media ISBN: 9783540307297 Category : Mathematics Languages : en Pages : 474
Book Description
One of the most remarkable and beautiful theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes and their connections with invariant theory, which has inspired hundreds of papers about generalizations and applications of this theorem to different types of codes. This self-contained book develops a new theory which is powerful enough to include all the earlier generalizations.
Author: W. Cary Huffman Publisher: CRC Press ISBN: 1351375105 Category : Computers Languages : en Pages : 998
Book Description
Most coding theory experts date the origin of the subject with the 1948 publication of A Mathematical Theory of Communication by Claude Shannon. Since then, coding theory has grown into a discipline with many practical applications (antennas, networks, memories), requiring various mathematical techniques, from commutative algebra, to semi-definite programming, to algebraic geometry. Most topics covered in the Concise Encyclopedia of Coding Theory are presented in short sections at an introductory level and progress from basic to advanced level, with definitions, examples, and many references. The book is divided into three parts: Part I fundamentals: cyclic codes, skew cyclic codes, quasi-cyclic codes, self-dual codes, codes and designs, codes over rings, convolutional codes, performance bounds Part II families: AG codes, group algebra codes, few-weight codes, Boolean function codes, codes over graphs Part III applications: alternative metrics, algorithmic techniques, interpolation decoding, pseudo-random sequences, lattices, quantum coding, space-time codes, network coding, distributed storage, secret-sharing, and code-based-cryptography. Features Suitable for students and researchers in a wide range of mathematical disciplines Contains many examples and references Most topics take the reader to the frontiers of research
Author: Clarice Dias de Albuquerque Publisher: Springer Nature ISBN: 3031068335 Category : Science Languages : en Pages : 123
Book Description
This book offers a structured algebraic and geometric approach to the classification and construction of quantum codes for topological quantum computation. It combines key concepts in linear algebra, algebraic topology, hyperbolic geometry, group theory, quantum mechanics, and classical and quantum coding theory to help readers understand and develop quantum codes for topological quantum computation. One possible approach to building a quantum computer is based on surface codes, operated as stabilizer codes. The surface codes evolved from Kitaev's toric codes, as a means to developing models for topological order by using qubits distributed on the surface of a toroid. A significant advantage of surface codes is their relative tolerance to local errors. A second approach is based on color codes, which are topological stabilizer codes defined on a tessellation with geometrically local stabilizer generators. This book provides basic geometric concepts, like surface geometry, hyperbolic geometry and tessellation, as well as basic algebraic concepts, like stabilizer formalism, for the construction of the most promising classes of quantum error-correcting codes such as surfaces codes and color codes. The book is intended for senior undergraduate and graduate students in Electrical Engineering and Mathematics with an understanding of the basic concepts of linear algebra and quantum mechanics.
Author: Reza Dastbasteh
Author: Reza Dastbasteh
Author: Santosh Kumar
Author: Avanti Ulhas Ketkar
Author: Tom Verhoeff
Author: Martin, W. J. (William Joseph)
Author: Martin Tomlinson
Author: Gabriele Nebe
Author: Homayoun Shahri
Author: W. Cary Huffman
Author: Clarice Dias de Albuquerque