00000cmm a22000004a 4500 UP-99796217611212520 Buklod 20230215085534.0 m go j cr |n |||auu|a 110823s2011 xxu u | eng d 9780817682569 (eBook) 0817682562 (eBook) (iLib)UPD-00215704313 DLC DML eng DMLUC QA 268 J69 2011eb Joyner, David. Selected unsolved problems in coding theory [electronic resource] David Joyner, Jon-Lark Kim. Boston Birkhäuser c2011. 1 online resource (xi, 203 p.) ill. Applied and Numerical Harmonic Analysis Preface -- Background -- Codes and Lattices -- Kittens and Blackjack -- RH and Coding Theory -- Hyperelliptic Curves and QR Codes -- Codes from Modular Curves -- Appendix -- Bibliography -- Index. IP-based subscription, access limited to within on campus computer network. Access via Electronic Resources of the UPD University Library Website. 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 transmitter 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 interseacts 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. Electronic reproduction. New York SpringerLink 2011. Available via World Wide Web through SpringerLink. Coding theory. Error-correcting codes (Information theory). Electronic books. Kim, Jon-Lark. SpringerLink (Online service). Electronic Resource Available for University of the Philippines Diliman via SpringerLink. Click here to access http://link.springer.com/book/10.1007/978-0-8176-8256-9 FO Monograph UPD DMLR Electronic Resource