Coding Theory with Methods from Algebraic Geometry and Number Theory

在以前的工作中，PI介绍了局部Artinian环上代数几何码的研究，这个项目的第一个主要目标是进一步理解这些对象。 特别地，研究者研究这些代码的最小平方欧几里德重量。 这个量与指数和密切相关，其中和是在代码的构造中使用的曲线上的点（在环上定义）。 因此，这些码的纠错能力问题被简化为数论中的一个问题。 结果已经获得了由调查员和一个同事在两种特殊情况下，其中所涉及的曲线要么是Serre-Tate正则提升的ordinaryelliptical曲线或平面曲线与一个单一的点在无穷远。 PI正在继续这项工作，目标是研究更一般曲线的平方欧几里德权重和相关指数和。 此外，PI正在努力开发这些代码的编码算法，关于平方欧几里德权重。 该项目的第二个主要目标涉及到代码的结构理论，一个几乎从代码研究开始就一直围绕的主题。 这种搜索基本上是不成功的，直到在最近的一篇论文中介绍了所谓的criticalindecomposable码的Assmus。PI正在进一步发展这一理论，并通过重新审视自对偶码的分类来证明它的威力。无论何时，数据通过信道传输，都必然会发生错误。 编码理论的目标是找到有效的方法来增加冗余，以便可以检测错误，甚至纠正错误。 对于每一个代码，必须问两个重要的问题：“这个代码的纠错能力是什么？如果在使用这个代码的过程中出现错误，有没有一种有效的方法来恢复原来的代码字？“在这种情况下，代码是定义在环的整数模幂的素数，theerror-correcting能力的代码是衡量方面的代码的最小平方欧几里德重量和研究人员studiesthis属性的情况下，代数几何码。 恢复原始码字的问题相当于为该码寻找一种编码算法，它与密码学也有着重要的联系。 最后，代码的结构理论允许从系统的角度研究代码。 特别是自对偶码往往在纠错能力和效率之间取得了很好的平衡，结构理论为研究这些对象提供了一种新的方法。

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.