Algebraic Geometric Codes over Rings

Whenever information is transmitted across a channel, errors are likely to occur. Since Shannon’s groundbreaking paper [36], coding theorists have sought to construct codes that have many codewords, that are easy to encode and decode, and that correct errors. While the main tools used in coding theory have traditionally been those of combinatorics and group theory, this volume is dedicated to codes constructed using algebraic geometry. Such codes were first introduced by Goppa [13] in 1977; see Definition 1.1 below. Soon after Goppa’s original paper, Tsfasman, Vlăduţ and Zink [43] used modular curves to construct a sequence of codes with asymptotically better parameters than any previously known codes. Thus, the study of algebraic geometric codes took on great significance. The field of coding theory took another major turn with the 1994 paper of Hammons, Kumar, Calderbank, Sloane and Solé [15] that shows that certain nonlinear binary codes are, in fact, nonlinear images of linear codes over the ring Z/4Z. The study of linear codes over rings has continued to mature into a mathematical field of study in its own right, causing Alexander Barg, Professor of Electrical and Computer Engineering at the