Coding Theory with Methods from Algebraic Geometry and Number Theory

点击翻译按钮获取中文摘要

In previous work, the PI introduced the study of algebraic geometriccodes over local Artinian rings, and the first main goal of thisproject is to further the understanding of these objects. Inparticular, the investigator studies the minimum squared Euclideanweight of these codes. This quantity is closely related to anexponential sum, where the sum is over points on the curve (definedover the ring) used in the construction of the code. Thus, thequestion of the error-correcting capability of these codes is reducedto a question in number theory. Results have already been obtained bythe investigator and a colleague in the two special cases where thecurve involved is either the Serre-Tate canonical lift of an ordinaryelliptic curve or a plane curve with a single point at infinity. ThePI is continuing this work with the goal of studying the squaredEuclidean weight and associated exponential sum for more generalcurves. Additionally, the PI is working towards the development of adecoding algorithm for these codes, with respect to the squaredEuclidean weight. The second main goal of the project involves thesearch for a structure theory of codes, a topic which has been aroundalmost from the beginning of the study of codes. This search waslargely unsuccessful until the introduction of so-called criticalindecomposable codes in a recent paper of Assmus. The PI is developingthis theory further and demonstrating its power by revisiting theclassification of self-dual codes.Whenever data is transmitted across a channel, errors are bound tooccur. The goal of coding theory is to find efficient ways of addingredundancy so that errors can be detected, or even corrected. Twobasic questions must be asked about every code: "What is theerror-correcting capability of this code?" and "If an error occurs intransmission while using this code, is there an efficient way ofrecovering the original codeword?" In the case that code is definedover the ring of integers modulo a power of a prime, theerror-correcting capability of the code is measured in terms of thecode's minimum squared Euclidean weight and the investigator studiesthis property in the case of algebraic geometric codes. The problemof recovering the original codeword is equivalent to finding adecoding algorithm for the code, which has important connections tocryptography also. Finally, a structure theory for codes allows thestudy of codes from a systematic point of view. In particular,self-dual codes often achieve a good balance between error-correctingcapability and efficiency, and a structure theory allows a new attackfor the study of these objects.