Coding Theory with Methods from Algebraic Geometry and Number Theory

在之前的工作中，PI介绍了局部阿提尼环上的代数几何码的研究，该项目的第一个主要目标是进一步了解这些对象。特别地，研究者研究了这些码的最小平方欧几里得权值。这个数量与指数和密切相关，其中和是在构建代码时使用的曲线（在环上定义）上的点。因此，这些码的纠错能力问题被简化为一个数论问题。研究者和他的同事已经在两种特殊情况下得到了结果，其中所涉及的曲线要么是普通椭圆曲线的sere - tate正则升，要么是在无穷远处有一个单点的平面曲线。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值和相关指数和。此外，PI正致力于开发这些代码的解码算法，考虑到平方欧几里得权重。该项目的第二个主要目标涉及对代码结构理论的研究，这一主题几乎从代码研究的开始就存在。这种搜索基本上是不成功的，直到在最近的一篇论文中引入了所谓的临界不可分解代码。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其力量。每当数据在信道上传输时，就必然会发生错误。编码理论的目标是找到增加冗余的有效方法，以便可以检测甚至纠正错误。对于每一个码，必须问两个基本问题：“这个码的纠错能力如何？”和“如果在使用这个码的传输过程中出现错误，是否有一种有效的方法可以恢复原始码字？”在这种情况下，代码是在整数环上定义的模一个素数的幂，代码的纠错能力是根据代码的最小平方欧几里得权来测量的，研究者研究了代数几何代码的这种性质。恢复原始码字的问题相当于找到编码的解码算法，这与密码学也有重要的联系。最后，语码的结构理论允许从系统的角度来研究语码。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新的突破口。

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.