Umfang:
Online-Ressource (XII, 248p. 17 illus, digital)
Ausgabe:
1
ISBN:
9780817682569
Serie:
Applied and Numerical Harmonic Analysis
Inhalt:
Preface -- Background -- Codes and Lattices -- Kittens and Blackjack -- RH and Coding Theory -- Hyperelliptic Curves and QR Codes -- Codes from Modular Curves -- Appendix -- Bibliography -- Index.
Inhalt:
Using an original mode of presentation and emphasizing the computational nature of the subject, this book explores a number of the unsolved problems that continue to exist in coding theory. A well-established and still highly relevant branch of mathematics, the theory of error-correcting codes is concerned with reliably transmitting data over a ‘noisy’ channel. Despite its frequent use in a range of contexts—the first close-up pictures of the surface of Mars, taken by the NASA spacecraft Mariner 9, were transmitted back to Earth using a Reed–Muller code—the subject contains interesting problems that have to date resisted solution by some of the most prominent mathematicians of recent decades. Employing SAGE—a free open-source mathematics software system—to illustrate their ideas, the authors begin by providing background on linear block codes and introducing some of the special families of codes explored in later chapters, such as quadratic residue and algebraic-geometric codes. Also surveyed is the theory that intersects self-dual codes, lattices, and invariant theory, which leads to an intriguing analogy between the Duursma zeta function and the zeta function attached to an algebraic curve over a finite field. The authors then examine a connection between the theory of block designs and the Assmus–Mattson theorem and scrutinize the knotty problem of finding a non-trivial estimate for the number of solutions over a finite field to a hyperelliptic polynomial equation of "small" degree, as well as the best asymptotic bounds for a binary linear block code. Finally, some of the more mysterious aspects relating modular forms and algebraic-geometric codes are discussed. Selected Unsolved Problems in Coding Theory is intended for graduate students and researchers in algebraic coding theory, especially those who are interested in finding current unsolved problems. Familiarity with concepts in algebra, number theory, and modular forms is assumed. The work may be used as supplementary reading material in a graduate course on coding theory or for self-study.
Anmerkung:
Includes bibliographical references and index
,
""Selected Unsolved Problems in Coding Theory""; ""Preface""; ""Acknowledgements""; ""Contents""; ""Chapter 1: Background on Information Theory and Coding Theory""; ""1.1 Binary Symmetric Channel""; ""1.1.1 Uncertainty""; ""1.1.2 Shannon's Theorem""; ""1.2 A Simple Example""; ""An Algorithm for Decoding the Hamming [7,4,3] Code""; ""1.3 Basic Definitions""; ""1.3.1 The Hamming Metric""; ""1.4 Linear Block Codes""; ""1.4.1 Decoding Basics""; ""1.4.2 Hamming Codes over GF(q)""; ""1.5 Bounds on the Parameters of a Code""; ""1.5.1 Question: What Is ""The Best"" Code?""
,
"" Asymptotic Bounds-Manin's Theorem"""" Asymptotic Bounds-The Plotkin Curve""; "" Asymptotic Bounds-The Gilbert-Varshamov Curve""; ""1.5.2 The Fake Singleton Bound""; ""1.6 Quadratic Residue Codes and Other Group Codes""; ""1.6.1 Automorphism Groups""; ""1.6.2 Cyclic Codes""; ""1.6.3 Quadratic Residue Codes""; "" Fourier Transforms on Finite Fields""; "" Generalized Quadratic Residue Codes""; "" Extended Quadratic Residue Codes""; ""Chapter 2: Self-dual Codes, Lattices, and Invariant Theory""; ""2.1 Weight Enumerators""; ""2.2 Divisible Codes""; ""2.3 Some Invariants""
,
""2.4 Codes over Other Finite Rings""""2.5 Lattices from Codes""; ""2.5.1 Constructions from Codes""; ""2.5.2 Theta Function of a Lattice""; ""2.6 More Problems Related to a Prize Problem""; ""Chapter 3: Kittens, Mathematical Blackjack, and Combinatorial Codes""; ""3.1 Hadamard Matrices and Codes""; ""3.2 Designs, Orthogonal Arrays, Latin Squares, and Codes""; ""3.2.1 Examples from Golay Codes""; ""3.2.2 Assmus-Mattson Theorem""; ""3.2.3 Orthogonal Arrays, Latin Squares and Codes""; ""3.3 Curtis' Kitten, Conway's Minimog""; ""3.3.1 The MINIMOG Description""
,
""3.3.2 Construction of the Extended Ternary Golay Code""""3.3.3 The ""col/tet"" Construction""; ""3.3.4 The Kitten Labeling""; ""3.4 Playing ""Mathematical Blackjack""""; ""3.5 Playing the Horses""; ""Chapter 4: The Riemann Hypothesis and Coding Theory""; ""4.1 Introduction to the Riemann Zeta Function""; ""4.2 Introduction to the Duursma Zeta Function""; ""4.3 Introduction""; ""4.3.1 Virtual Weight Enumerators""; ""4.4 The Zeta Polynomial""; ""4.4.1 First Definition""; ""4.4.2 Second Definition""; ""4.4.3 Third Definition""; ""4.4.4 Analogies with Curves""; ""4.5 Properties""
,
""4.5.1 The Functional Equation""""4.5.2 Puncturing Preserves P""; ""4.5.3 The Riemann Hypothesis""; ""4.6 Self-reciprocal Polynomials""; ""4.6.1 ""Smoothness"" of Roots""; ""4.6.2 Variations on a Theorem of Eneström-Kakeya""; ""4.6.3 A Literature Survey""; ""4.6.4 Duursma's Conjecture""; ""4.6.5 A Conjecture on Zeros of Cosine Transforms""; ""4.7 Examples""; ""4.7.1 Komichi's Example""; ""4.7.2 The Extremal Case""; ""4.7.3 ""Random Divisible Codes""""; ""4.7.4 A Formally Self-dual [26,13,6]2-code""; ""4.7.5 Extremal Codes of Short Length""; ""4.7.6 Non-self-dual Examples""
,
""4.8 Chinen Zeta Functions""
Weitere Ausg.:
ISBN 9780817682552
Weitere Ausg.:
Buchausg. u.d.T. Joyner, David, 1959 - Selected unsolved problems in coding theory New York : Birkhauser, 2011 ISBN 0817682554
Weitere Ausg.:
ISBN 9780817682552
Sprache:
Englisch
Fachgebiete:
Technik
,
Mathematik
Schlagwort(e):
Codierungstheorie
;
Codierungstheorie
;
Bibliografie
DOI:
10.1007/978-0-8176-8256-9
URL:
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