Algebraic Coding Theory

This book provides a self-contained introduction to algebraic coding theory over finite Frobenius rings. It is the first to offer a comprehensive account on the subject.

The last twenty-fit,e years have witnessed thr: growth of one of the most elegant and esoteric branches of applied mathematics: Algebraic Coding Theory. Areas of mathematics which were previously considered to be of the utmost purity have been applied to the problem of constructing error-correcting codes and their decoding algorithms. In spite of the impressive theoretical accomplishments of these twenty-five years, however, only recently has algebraic coding been put into practice.

The last twenty-fit,e years have witnessed thr: growth of one of the most elegant and esoteric branches of applied mathematics: Algebraic Coding Theory. Areas of mathematics which were previously considered to be of the utmost purity have been applied to the problem of constructing error-correcting codes and their decoding algorithms. In spite of the impressive theoretical accomplishments of these twenty-five years, however, only recently has algebraic coding been put into practice.

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.

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.

These are the proceedings of the Conference on Coding Theory, Cryptography, and Number Theory held at the U. S. Naval Academy during October 25-26, 1998. This book concerns elementary and advanced aspects of coding theory and cryptography. The coding theory contributions deal mostly with algebraic coding theory. Some of these papers are expository, whereas others are the result of original research. The emphasis is on geometric Goppa codes (Shokrollahi, Shokranian-Joyner), but there is also a paper on codes arising from combinatorial constructions (Michael). There are both, historical and mathematical papers on cryptography. Several of the contributions on cryptography describe the work done by the British and their allies during World War II to crack the German and Japanese ciphers (Hamer, Hilton, Tutte, Weierud, Urling). Some mathematical aspects of the Enigma rotor machine (Sherman) and more recent research on quantum cryptography (Lomonoco) are described. There are two papers concerned with the RSA cryptosystem and related number-theoretic issues (Wardlaw, Cosgrave).

These notes are based on lectures given in the semmar on "Coding Theory and Algebraic Geometry" held at Schloss Mickeln, Diisseldorf, November 16-21, 1987. In 1982 Tsfasman, Vladut and Zink, using algebraic geometry and ideas of Goppa, constructed a seqeunce of codes that exceed the Gilbert-Varshamov bound. The result was considered sensational.

This book provides a self-contained introduction to algebraic coding theory over finite Frobenius rings. It is the first to offer a comprehensive account on the subject.