9780817682569_

Using an original mode of presentation, and emphasizing the computational nature of the subject, this book explores a number of the unsolved problems that still exist in coding theory. A well-established, yet 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 contextsthe first close-up pictures of the surface of Mars, taken by the NASA spacecraft Mariner 9, were transmitted back to Earth using a ReedMuller codethe subject still contains interesting unsolved problems that have resisted solution by some of the most prominent mathematicians of recent decades.

Employing Sagea free open-source mathematics software systemto 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 with the theory of block designs and the AssmusMattson theorem. Further chapters 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. Some of the more mysterious aspects relating modular forms and algebraic-geometric codes are also 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 some 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.

(*_*)9780817682569_

This original monograph investigates several unsolved problems that currently exist in coding theory. A highly relevant branch of mathematical computer science, the theory of error-correcting codes is concerned with reliably transmitting data over a noisy channel. Despite its fairly long history and consistent prominence, the field still contains interesting problems that have resisted solution by some of the most prominent mathematicians of recent decades.

Employing SAGEa free open-source mathematics software systemto illustrate ideas, this book is intended for graduate students and researchers in algebraic coding theory, especially those who are interested in finding some 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.

...

Filtros

6e9254c6-8a93-3414-93da-3d65a0eb71bf